diff --git a/compiler/rustc_infer/src/infer/combine.rs b/compiler/rustc_infer/src/infer/combine.rs
index dfae279324f..cfd53fc71eb 100644
--- a/compiler/rustc_infer/src/infer/combine.rs
+++ b/compiler/rustc_infer/src/infer/combine.rs
@@ -34,8 +34,8 @@ use rustc_middle::infer::unify_key::{ConstVarValue, ConstVariableValue, EffectVa
 use rustc_middle::infer::unify_key::{ConstVariableOrigin, ConstVariableOriginKind};
 use rustc_middle::ty::error::{ExpectedFound, TypeError};
 use rustc_middle::ty::relate::{RelateResult, TypeRelation};
-use rustc_middle::ty::TyVar;
 use rustc_middle::ty::{self, InferConst, ToPredicate, Ty, TyCtxt, TypeVisitableExt};
+use rustc_middle::ty::{AliasRelationDirection, TyVar};
 use rustc_middle::ty::{IntType, UintType};
 use rustc_span::DUMMY_SP;
 
@@ -490,31 +490,47 @@ impl<'infcx, 'tcx> CombineFields<'infcx, 'tcx> {
             // cyclic type. We instead delay the unification in case
             // the alias can be normalized to something which does not
             // mention `?0`.
-
-            // FIXME(-Ztrait-solver=next): replace this with `AliasRelate`
-            let &ty::Alias(kind, data) = a_ty.kind() else {
-                bug!("generalization should only result in infer vars for aliases");
-            };
-            if !self.infcx.next_trait_solver() {
-                // The old solver only accepts projection predicates for associated types.
-                match kind {
-                    ty::AliasKind::Projection => {}
-                    ty::AliasKind::Inherent | ty::AliasKind::Weak | ty::AliasKind::Opaque => {
-                        return Err(TypeError::CyclicTy(a_ty));
+            if self.infcx.next_trait_solver() {
+                let (lhs, rhs, direction) = match ambient_variance {
+                    ty::Variance::Invariant => {
+                        (a_ty.into(), b_ty.into(), AliasRelationDirection::Equate)
                     }
+                    ty::Variance::Covariant => {
+                        (a_ty.into(), b_ty.into(), AliasRelationDirection::Subtype)
+                    }
+                    ty::Variance::Contravariant => {
+                        (b_ty.into(), a_ty.into(), AliasRelationDirection::Subtype)
+                    }
+                    ty::Variance::Bivariant => unreachable!("bivariant generalization"),
+                };
+                self.obligations.push(Obligation::new(
+                    self.tcx(),
+                    self.trace.cause.clone(),
+                    self.param_env,
+                    ty::PredicateKind::AliasRelate(lhs, rhs, direction),
+                ));
+            } else {
+                match a_ty.kind() {
+                    &ty::Alias(ty::AliasKind::Projection, data) => {
+                        // FIXME: This does not handle subtyping correctly, we should switch to
+                        // alias-relate in the new solver and could instead create a new inference
+                        // variable for `a_ty`, emitting `Projection(a_ty, a_infer)` and
+                        // `a_infer <: b_ty`.
+                        self.obligations.push(Obligation::new(
+                            self.tcx(),
+                            self.trace.cause.clone(),
+                            self.param_env,
+                            ty::ProjectionPredicate { projection_ty: data, term: b_ty.into() },
+                        ))
+                    }
+                    // The old solver only accepts projection predicates for associated types.
+                    ty::Alias(
+                        ty::AliasKind::Inherent | ty::AliasKind::Weak | ty::AliasKind::Opaque,
+                        _,
+                    ) => return Err(TypeError::CyclicTy(a_ty)),
+                    _ => bug!("generalizated `{a_ty:?} to infer, not an alias"),
                 }
             }
-
-            // FIXME: This does not handle subtyping correctly, we should switch to
-            // alias-relate in the new solver and could instead create a new inference
-            // variable for `a_ty`, emitting `Projection(a_ty, a_infer)` and
-            // `a_infer <: b_ty`.
-            self.obligations.push(Obligation::new(
-                self.tcx(),
-                self.trace.cause.clone(),
-                self.param_env,
-                ty::ProjectionPredicate { projection_ty: data, term: b_ty.into() },
-            ))
         } else {
             match ambient_variance {
                 ty::Variance::Invariant => self.equate(a_is_expected).relate(a_ty, b_ty),
@@ -525,9 +541,7 @@ impl<'infcx, 'tcx> CombineFields<'infcx, 'tcx> {
                     a_ty,
                     b_ty,
                 ),
-                ty::Variance::Bivariant => {
-                    unreachable!("no code should be generalizing bivariantly (currently)")
-                }
+                ty::Variance::Bivariant => unreachable!("bivariant generalization"),
             }?;
         }
 
diff --git a/compiler/rustc_middle/src/ty/print/pretty.rs b/compiler/rustc_middle/src/ty/print/pretty.rs
index faf2a854f69..57253fa7c59 100644
--- a/compiler/rustc_middle/src/ty/print/pretty.rs
+++ b/compiler/rustc_middle/src/ty/print/pretty.rs
@@ -2814,7 +2814,7 @@ define_print! {
                 p!("the constant `", print(c1), "` equals `", print(c2), "`")
             }
             ty::PredicateKind::Ambiguous => p!("ambiguous"),
-            ty::PredicateKind::NormalizesTo(data) => p!(print(data.alias), " normalizes-to ", print(data.term)),
+            ty::PredicateKind::NormalizesTo(data) => p!(print(data)),
             ty::PredicateKind::AliasRelate(t1, t2, dir) => p!(print(t1), write(" {} ", dir), print(t2)),
         }
     }
@@ -2946,6 +2946,12 @@ define_print_and_forward_display! {
         p!(print(self.term))
     }
 
+    ty::NormalizesTo<'tcx> {
+        p!(print(self.alias), " normalizes-to ");
+        cx.reset_type_limit();
+        p!(print(self.term))
+    }
+
     ty::Term<'tcx> {
       match self.unpack() {
         ty::TermKind::Ty(ty) => p!(print(ty)),
diff --git a/compiler/rustc_trait_selection/src/solve/alias_relate.rs b/compiler/rustc_trait_selection/src/solve/alias_relate.rs
index 739bbe929b3..2e99854ddc6 100644
--- a/compiler/rustc_trait_selection/src/solve/alias_relate.rs
+++ b/compiler/rustc_trait_selection/src/solve/alias_relate.rs
@@ -92,7 +92,7 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
                     self.add_goal(Goal::new(
                         self.tcx(),
                         param_env,
-                        ty::ProjectionPredicate { projection_ty: alias, term },
+                        ty::NormalizesTo { alias, term },
                     ));
                     self.try_evaluate_added_goals()?;
                     Ok(Some(self.resolve_vars_if_possible(term)))
@@ -109,11 +109,7 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
         opaque: ty::AliasTy<'tcx>,
         term: ty::Term<'tcx>,
     ) -> QueryResult<'tcx> {
-        self.add_goal(Goal::new(
-            self.tcx(),
-            param_env,
-            ty::ProjectionPredicate { projection_ty: opaque, term },
-        ));
+        self.add_goal(Goal::new(self.tcx(), param_env, ty::NormalizesTo { alias: opaque, term }));
         self.evaluate_added_goals_and_make_canonical_response(Certainty::Yes)
     }
 
diff --git a/compiler/rustc_trait_selection/src/solve/assembly/mod.rs b/compiler/rustc_trait_selection/src/solve/assembly/mod.rs
index b6861d258d1..201fade5ad7 100644
--- a/compiler/rustc_trait_selection/src/solve/assembly/mod.rs
+++ b/compiler/rustc_trait_selection/src/solve/assembly/mod.rs
@@ -352,15 +352,13 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
         num_steps: usize,
     ) {
         let tcx = self.tcx();
-        let &ty::Alias(_, projection_ty) = goal.predicate.self_ty().kind() else { return };
+        let &ty::Alias(_, alias) = goal.predicate.self_ty().kind() else { return };
 
         candidates.extend(self.probe(|_| ProbeKind::NormalizedSelfTyAssembly).enter(|ecx| {
             if tcx.recursion_limit().value_within_limit(num_steps) {
                 let normalized_ty = ecx.next_ty_infer();
-                let normalizes_to_goal = goal.with(
-                    tcx,
-                    ty::ProjectionPredicate { projection_ty, term: normalized_ty.into() },
-                );
+                let normalizes_to_goal =
+                    goal.with(tcx, ty::NormalizesTo { alias, term: normalized_ty.into() });
                 ecx.add_goal(normalizes_to_goal);
                 if let Err(NoSolution) = ecx.try_evaluate_added_goals() {
                     debug!("self type normalization failed");
diff --git a/compiler/rustc_trait_selection/src/solve/canonicalize.rs b/compiler/rustc_trait_selection/src/solve/canonicalize.rs
index 004dc45263c..469d1f5a9cb 100644
--- a/compiler/rustc_trait_selection/src/solve/canonicalize.rs
+++ b/compiler/rustc_trait_selection/src/solve/canonicalize.rs
@@ -69,8 +69,8 @@ impl<'a, 'tcx> Canonicalizer<'a, 'tcx> {
         };
 
         let value = value.fold_with(&mut canonicalizer);
-        assert!(!value.has_infer());
-        assert!(!value.has_placeholders());
+        assert!(!value.has_infer(), "unexpected infer in {value:?}");
+        assert!(!value.has_placeholders(), "unexpected placeholders in {value:?}");
 
         let (max_universe, variables) = canonicalizer.finalize();
 
diff --git a/compiler/rustc_trait_selection/src/solve/eval_ctxt/mod.rs b/compiler/rustc_trait_selection/src/solve/eval_ctxt/mod.rs
index 348dfdf725f..b3e7a63c972 100644
--- a/compiler/rustc_trait_selection/src/solve/eval_ctxt/mod.rs
+++ b/compiler/rustc_trait_selection/src/solve/eval_ctxt/mod.rs
@@ -103,7 +103,7 @@ pub(super) struct NestedGoals<'tcx> {
     /// with a fresh inference variable when we evaluate this goal. That can result
     /// in a trait solver cycle. This would currently result in overflow but can be
     /// can be unsound with more powerful coinduction in the future.
-    pub(super) normalizes_to_hack_goal: Option<Goal<'tcx, ty::ProjectionPredicate<'tcx>>>,
+    pub(super) normalizes_to_hack_goal: Option<Goal<'tcx, ty::NormalizesTo<'tcx>>>,
     /// The rest of the goals which have not yet processed or remain ambiguous.
     pub(super) goals: Vec<Goal<'tcx, ty::Predicate<'tcx>>>,
 }
@@ -423,7 +423,9 @@ impl<'a, 'tcx> EvalCtxt<'a, 'tcx> {
                 ty::PredicateKind::ConstEquate(_, _) => {
                     bug!("ConstEquate should not be emitted when `-Ztrait-solver=next` is active")
                 }
-                ty::PredicateKind::NormalizesTo(_) => unimplemented!(),
+                ty::PredicateKind::NormalizesTo(predicate) => {
+                    self.compute_normalizes_to_goal(Goal { param_env, predicate })
+                }
                 ty::PredicateKind::AliasRelate(lhs, rhs, direction) => self
                     .compute_alias_relate_goal(Goal {
                         param_env,
@@ -493,10 +495,7 @@ impl<'a, 'tcx> EvalCtxt<'a, 'tcx> {
             let unconstrained_rhs = self.next_term_infer_of_kind(goal.predicate.term);
             let unconstrained_goal = goal.with(
                 tcx,
-                ty::ProjectionPredicate {
-                    projection_ty: goal.predicate.projection_ty,
-                    term: unconstrained_rhs,
-                },
+                ty::NormalizesTo { alias: goal.predicate.alias, term: unconstrained_rhs },
             );
 
             let (_, certainty, instantiate_goals) = self.evaluate_goal(
@@ -518,9 +517,7 @@ impl<'a, 'tcx> EvalCtxt<'a, 'tcx> {
             // looking at the "has changed" return from evaluate_goal,
             // because we expect the `unconstrained_rhs` part of the predicate
             // to have changed -- that means we actually normalized successfully!
-            if goal.predicate.projection_ty
-                != self.resolve_vars_if_possible(goal.predicate.projection_ty)
-            {
+            if goal.predicate.alias != self.resolve_vars_if_possible(goal.predicate.alias) {
                 unchanged_certainty = None;
             }
 
@@ -590,9 +587,10 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
     ///
     /// This is the case if the `term` is an inference variable in the innermost universe
     /// and does not occur in any other part of the predicate.
+    #[instrument(level = "debug", skip(self), ret)]
     pub(super) fn term_is_fully_unconstrained(
         &self,
-        goal: Goal<'tcx, ty::ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, ty::NormalizesTo<'tcx>>,
     ) -> bool {
         let term_is_infer = match goal.predicate.term.unpack() {
             ty::TermKind::Ty(ty) => {
@@ -656,7 +654,7 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
         let mut visitor = ContainsTerm { infcx: self.infcx, term: goal.predicate.term };
 
         term_is_infer
-            && goal.predicate.projection_ty.visit_with(&mut visitor).is_continue()
+            && goal.predicate.alias.visit_with(&mut visitor).is_continue()
             && goal.param_env.visit_with(&mut visitor).is_continue()
     }
 
diff --git a/compiler/rustc_trait_selection/src/solve/mod.rs b/compiler/rustc_trait_selection/src/solve/mod.rs
index bf3b72caeb4..cf3f94e26e4 100644
--- a/compiler/rustc_trait_selection/src/solve/mod.rs
+++ b/compiler/rustc_trait_selection/src/solve/mod.rs
@@ -35,6 +35,7 @@ mod eval_ctxt;
 mod fulfill;
 pub mod inspect;
 mod normalize;
+mod normalizes_to;
 mod project_goals;
 mod search_graph;
 mod trait_goals;
@@ -216,7 +217,7 @@ impl<'a, 'tcx> EvalCtxt<'a, 'tcx> {
 
 impl<'tcx> EvalCtxt<'_, 'tcx> {
     #[instrument(level = "debug", skip(self))]
-    fn set_normalizes_to_hack_goal(&mut self, goal: Goal<'tcx, ty::ProjectionPredicate<'tcx>>) {
+    fn set_normalizes_to_hack_goal(&mut self, goal: Goal<'tcx, ty::NormalizesTo<'tcx>>) {
         assert!(
             self.nested_goals.normalizes_to_hack_goal.is_none(),
             "attempted to set the projection eq hack goal when one already exists"
@@ -310,17 +311,17 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
             return None;
         }
 
-        let ty::Alias(kind, projection_ty) = *ty.kind() else {
+        let ty::Alias(kind, alias) = *ty.kind() else {
             return Some(ty);
         };
 
         // We do no always define opaque types eagerly to allow non-defining uses in the defining scope.
         if let (DefineOpaqueTypes::No, ty::AliasKind::Opaque) = (define_opaque_types, kind) {
-            if let Some(def_id) = projection_ty.def_id.as_local() {
+            if let Some(def_id) = alias.def_id.as_local() {
                 if self
                     .unify_existing_opaque_tys(
                         param_env,
-                        OpaqueTypeKey { def_id, args: projection_ty.args },
+                        OpaqueTypeKey { def_id, args: alias.args },
                         self.next_ty_infer(),
                     )
                     .is_empty()
@@ -335,7 +336,7 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
             let normalizes_to_goal = Goal::new(
                 this.tcx(),
                 param_env,
-                ty::ProjectionPredicate { projection_ty, term: normalized_ty.into() },
+                ty::NormalizesTo { alias, term: normalized_ty.into() },
             );
             this.add_goal(normalizes_to_goal);
             this.try_evaluate_added_goals()?;
diff --git a/compiler/rustc_trait_selection/src/solve/normalize.rs b/compiler/rustc_trait_selection/src/solve/normalize.rs
index ea512ba5fa7..1e495b4d979 100644
--- a/compiler/rustc_trait_selection/src/solve/normalize.rs
+++ b/compiler/rustc_trait_selection/src/solve/normalize.rs
@@ -76,7 +76,7 @@ impl<'tcx> NormalizationFolder<'_, 'tcx> {
             tcx,
             self.at.cause.clone(),
             self.at.param_env,
-            ty::ProjectionPredicate { projection_ty: alias, term: new_infer_ty.into() },
+            ty::NormalizesTo { alias, term: new_infer_ty.into() },
         );
 
         // Do not emit an error if normalization is known to fail but instead
@@ -129,8 +129,8 @@ impl<'tcx> NormalizationFolder<'_, 'tcx> {
             tcx,
             self.at.cause.clone(),
             self.at.param_env,
-            ty::ProjectionPredicate {
-                projection_ty: AliasTy::new(tcx, uv.def, uv.args),
+            ty::NormalizesTo {
+                alias: AliasTy::new(tcx, uv.def, uv.args),
                 term: new_infer_ct.into(),
             },
         );
diff --git a/compiler/rustc_trait_selection/src/solve/project_goals/inherent_projection.rs b/compiler/rustc_trait_selection/src/solve/normalizes_to/inherent.rs
similarity index 94%
rename from compiler/rustc_trait_selection/src/solve/project_goals/inherent_projection.rs
rename to compiler/rustc_trait_selection/src/solve/normalizes_to/inherent.rs
index 28fe59b7f6a..c3b8ae9a943 100644
--- a/compiler/rustc_trait_selection/src/solve/project_goals/inherent_projection.rs
+++ b/compiler/rustc_trait_selection/src/solve/normalizes_to/inherent.rs
@@ -12,10 +12,10 @@ use super::EvalCtxt;
 impl<'tcx> EvalCtxt<'_, 'tcx> {
     pub(super) fn normalize_inherent_associated_type(
         &mut self,
-        goal: Goal<'tcx, ty::ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, ty::NormalizesTo<'tcx>>,
     ) -> QueryResult<'tcx> {
         let tcx = self.tcx();
-        let inherent = goal.predicate.projection_ty;
+        let inherent = goal.predicate.alias;
         let expected = goal.predicate.term.ty().expect("inherent consts are treated separately");
 
         let impl_def_id = tcx.parent(inherent.def_id);
diff --git a/compiler/rustc_trait_selection/src/solve/project_goals/mod.rs b/compiler/rustc_trait_selection/src/solve/normalizes_to/mod.rs
similarity index 95%
rename from compiler/rustc_trait_selection/src/solve/project_goals/mod.rs
rename to compiler/rustc_trait_selection/src/solve/normalizes_to/mod.rs
index 03823569669..867a520915f 100644
--- a/compiler/rustc_trait_selection/src/solve/project_goals/mod.rs
+++ b/compiler/rustc_trait_selection/src/solve/normalizes_to/mod.rs
@@ -13,20 +13,20 @@ use rustc_middle::traits::solve::{
 };
 use rustc_middle::traits::BuiltinImplSource;
 use rustc_middle::ty::fast_reject::{DeepRejectCtxt, TreatParams};
-use rustc_middle::ty::ProjectionPredicate;
+use rustc_middle::ty::NormalizesTo;
 use rustc_middle::ty::{self, Ty, TyCtxt};
 use rustc_middle::ty::{ToPredicate, TypeVisitableExt};
 use rustc_span::{sym, ErrorGuaranteed, DUMMY_SP};
 
-mod inherent_projection;
+mod inherent;
 mod opaques;
 mod weak_types;
 
 impl<'tcx> EvalCtxt<'_, 'tcx> {
     #[instrument(level = "debug", skip(self), ret)]
-    pub(super) fn compute_projection_goal(
+    pub(super) fn compute_normalizes_to_goal(
         &mut self,
-        goal: Goal<'tcx, ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, NormalizesTo<'tcx>>,
     ) -> QueryResult<'tcx> {
         let def_id = goal.predicate.def_id();
         match self.tcx().def_kind(def_id) {
@@ -71,16 +71,13 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
     #[instrument(level = "debug", skip(self), ret)]
     fn normalize_anon_const(
         &mut self,
-        goal: Goal<'tcx, ty::ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, ty::NormalizesTo<'tcx>>,
     ) -> QueryResult<'tcx> {
         if let Some(normalized_const) = self.try_const_eval_resolve(
             goal.param_env,
-            ty::UnevaluatedConst::new(
-                goal.predicate.projection_ty.def_id,
-                goal.predicate.projection_ty.args,
-            ),
+            ty::UnevaluatedConst::new(goal.predicate.alias.def_id, goal.predicate.alias.args),
             self.tcx()
-                .type_of(goal.predicate.projection_ty.def_id)
+                .type_of(goal.predicate.alias.def_id)
                 .no_bound_vars()
                 .expect("const ty should not rely on other generics"),
         ) {
@@ -92,13 +89,13 @@ impl<'tcx> EvalCtxt<'_, 'tcx> {
     }
 }
 
-impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
+impl<'tcx> assembly::GoalKind<'tcx> for NormalizesTo<'tcx> {
     fn self_ty(self) -> Ty<'tcx> {
         self.self_ty()
     }
 
     fn trait_ref(self, tcx: TyCtxt<'tcx>) -> ty::TraitRef<'tcx> {
-        self.projection_ty.trait_ref(tcx)
+        self.alias.trait_ref(tcx)
     }
 
     fn with_self_ty(self, tcx: TyCtxt<'tcx>, self_ty: Ty<'tcx>) -> Self {
@@ -123,7 +120,7 @@ impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
                         ecx.instantiate_binder_with_infer(projection_pred);
                     ecx.eq(
                         goal.param_env,
-                        goal.predicate.projection_ty,
+                        goal.predicate.alias,
                         assumption_projection_pred.projection_ty,
                     )?;
                     ecx.eq(goal.param_env, goal.predicate.term, assumption_projection_pred.term)
@@ -132,7 +129,7 @@ impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
                     // Add GAT where clauses from the trait's definition
                     ecx.add_goals(
                         tcx.predicates_of(goal.predicate.def_id())
-                            .instantiate_own(tcx, goal.predicate.projection_ty.args)
+                            .instantiate_own(tcx, goal.predicate.alias.args)
                             .map(|(pred, _)| goal.with(tcx, pred)),
                     );
 
@@ -148,12 +145,12 @@ impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
 
     fn consider_impl_candidate(
         ecx: &mut EvalCtxt<'_, 'tcx>,
-        goal: Goal<'tcx, ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, NormalizesTo<'tcx>>,
         impl_def_id: DefId,
     ) -> Result<Candidate<'tcx>, NoSolution> {
         let tcx = ecx.tcx();
 
-        let goal_trait_ref = goal.predicate.projection_ty.trait_ref(tcx);
+        let goal_trait_ref = goal.predicate.alias.trait_ref(tcx);
         let impl_trait_ref = tcx.impl_trait_ref(impl_def_id).unwrap();
         let drcx = DeepRejectCtxt { treat_obligation_params: TreatParams::ForLookup };
         if !drcx.args_may_unify(goal_trait_ref.args, impl_trait_ref.skip_binder().args) {
@@ -177,7 +174,7 @@ impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
             // Add GAT where clauses from the trait's definition
             ecx.add_goals(
                 tcx.predicates_of(goal.predicate.def_id())
-                    .instantiate_own(tcx, goal.predicate.projection_ty.args)
+                    .instantiate_own(tcx, goal.predicate.alias.args)
                     .map(|(pred, _)| goal.with(tcx, pred)),
             );
 
@@ -202,7 +199,7 @@ impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
                         tcx,
                         guar,
                         tcx.type_of(goal.predicate.def_id())
-                            .instantiate(tcx, goal.predicate.projection_ty.args),
+                            .instantiate(tcx, goal.predicate.alias.args),
                     )
                     .into(),
                     ty::AssocKind::Type => Ty::new_error(tcx, guar).into(),
@@ -227,11 +224,8 @@ impl<'tcx> assembly::GoalKind<'tcx> for ProjectionPredicate<'tcx> {
             //
             // And then map these args to the args of the defining impl of `Assoc`, going
             // from `[u32, u64]` to `[u32, i32, u64]`.
-            let impl_args_with_gat = goal.predicate.projection_ty.args.rebase_onto(
-                tcx,
-                goal_trait_ref.def_id,
-                impl_args,
-            );
+            let impl_args_with_gat =
+                goal.predicate.alias.args.rebase_onto(tcx, goal_trait_ref.def_id, impl_args);
             let args = ecx.translate_args(
                 goal.param_env,
                 impl_def_id,
diff --git a/compiler/rustc_trait_selection/src/solve/project_goals/opaques.rs b/compiler/rustc_trait_selection/src/solve/normalizes_to/opaques.rs
similarity index 97%
rename from compiler/rustc_trait_selection/src/solve/project_goals/opaques.rs
rename to compiler/rustc_trait_selection/src/solve/normalizes_to/opaques.rs
index 1fde129c3a0..b5d1aa06e4e 100644
--- a/compiler/rustc_trait_selection/src/solve/project_goals/opaques.rs
+++ b/compiler/rustc_trait_selection/src/solve/normalizes_to/opaques.rs
@@ -12,10 +12,10 @@ use crate::solve::{EvalCtxt, SolverMode};
 impl<'tcx> EvalCtxt<'_, 'tcx> {
     pub(super) fn normalize_opaque_type(
         &mut self,
-        goal: Goal<'tcx, ty::ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, ty::NormalizesTo<'tcx>>,
     ) -> QueryResult<'tcx> {
         let tcx = self.tcx();
-        let opaque_ty = goal.predicate.projection_ty;
+        let opaque_ty = goal.predicate.alias;
         let expected = goal.predicate.term.ty().expect("no such thing as an opaque const");
 
         match (goal.param_env.reveal(), self.solver_mode()) {
diff --git a/compiler/rustc_trait_selection/src/solve/project_goals/weak_types.rs b/compiler/rustc_trait_selection/src/solve/normalizes_to/weak_types.rs
similarity index 91%
rename from compiler/rustc_trait_selection/src/solve/project_goals/weak_types.rs
rename to compiler/rustc_trait_selection/src/solve/normalizes_to/weak_types.rs
index 54de32cf618..8d2bbec6d8b 100644
--- a/compiler/rustc_trait_selection/src/solve/project_goals/weak_types.rs
+++ b/compiler/rustc_trait_selection/src/solve/normalizes_to/weak_types.rs
@@ -11,10 +11,10 @@ use super::EvalCtxt;
 impl<'tcx> EvalCtxt<'_, 'tcx> {
     pub(super) fn normalize_weak_type(
         &mut self,
-        goal: Goal<'tcx, ty::ProjectionPredicate<'tcx>>,
+        goal: Goal<'tcx, ty::NormalizesTo<'tcx>>,
     ) -> QueryResult<'tcx> {
         let tcx = self.tcx();
-        let weak_ty = goal.predicate.projection_ty;
+        let weak_ty = goal.predicate.alias;
         let expected = goal.predicate.term.ty().expect("no such thing as a const alias");
 
         let actual = tcx.type_of(weak_ty.def_id).instantiate(tcx, weak_ty.args);
diff --git a/compiler/rustc_trait_selection/src/solve/project_goals.rs b/compiler/rustc_trait_selection/src/solve/project_goals.rs
new file mode 100644
index 00000000000..0b80969c307
--- /dev/null
+++ b/compiler/rustc_trait_selection/src/solve/project_goals.rs
@@ -0,0 +1,23 @@
+use super::EvalCtxt;
+use rustc_middle::traits::solve::{Certainty, Goal, QueryResult};
+use rustc_middle::ty::{self, ProjectionPredicate};
+
+impl<'tcx> EvalCtxt<'_, 'tcx> {
+    #[instrument(level = "debug", skip(self), ret)]
+    pub(super) fn compute_projection_goal(
+        &mut self,
+        goal: Goal<'tcx, ProjectionPredicate<'tcx>>,
+    ) -> QueryResult<'tcx> {
+        match goal.predicate.term.unpack() {
+            ty::TermKind::Ty(term) => {
+                let alias = goal.predicate.projection_ty.to_ty(self.tcx());
+                self.eq(goal.param_env, alias, term)?;
+                self.evaluate_added_goals_and_make_canonical_response(Certainty::Yes)
+            }
+            // FIXME(associated_const_equality): actually do something here.
+            ty::TermKind::Const(_) => {
+                self.evaluate_added_goals_and_make_canonical_response(Certainty::Yes)
+            }
+        }
+    }
+}
diff --git a/compiler/rustc_trait_selection/src/traits/structural_normalize.rs b/compiler/rustc_trait_selection/src/traits/structural_normalize.rs
index 9d6be768901..32de8feda81 100644
--- a/compiler/rustc_trait_selection/src/traits/structural_normalize.rs
+++ b/compiler/rustc_trait_selection/src/traits/structural_normalize.rs
@@ -25,8 +25,7 @@ impl<'tcx> StructurallyNormalizeExt<'tcx> for At<'_, 'tcx> {
             // FIXME(-Ztrait-solver=next): correctly handle
             // overflow here.
             for _ in 0..256 {
-                let ty::Alias(ty::Projection | ty::Inherent | ty::Weak, projection_ty) = *ty.kind()
-                else {
+                let ty::Alias(ty::Projection | ty::Inherent | ty::Weak, alias) = *ty.kind() else {
                     break;
                 };
 
@@ -38,10 +37,7 @@ impl<'tcx> StructurallyNormalizeExt<'tcx> for At<'_, 'tcx> {
                     self.infcx.tcx,
                     self.cause.clone(),
                     self.param_env,
-                    ty::Binder::dummy(ty::ProjectionPredicate {
-                        projection_ty,
-                        term: new_infer_ty.into(),
-                    }),
+                    ty::NormalizesTo { alias, term: new_infer_ty.into() },
                 );
                 if self.infcx.predicate_may_hold(&obligation) {
                     fulfill_cx.register_predicate_obligation(self.infcx, obligation);
diff --git a/compiler/rustc_type_ir/src/predicate_kind.rs b/compiler/rustc_type_ir/src/predicate_kind.rs
index adeb3f57497..b567fa8e2f6 100644
--- a/compiler/rustc_type_ir/src/predicate_kind.rs
+++ b/compiler/rustc_type_ir/src/predicate_kind.rs
@@ -120,7 +120,7 @@ where
 }
 
 #[derive(derivative::Derivative)]
-#[derivative(Clone(bound = ""), Hash(bound = ""))]
+#[derivative(Clone(bound = ""), Hash(bound = ""), PartialEq(bound = ""), Eq(bound = ""))]
 #[cfg_attr(feature = "nightly", derive(TyEncodable, TyDecodable, HashStable_NoContext))]
 pub enum PredicateKind<I: Interner> {
     /// Prove a clause
@@ -169,7 +169,6 @@ pub enum PredicateKind<I: Interner> {
     AliasRelate(I::Term, I::Term, AliasRelationDirection),
 }
 
-/// FIXME: This impl sh
 impl<I: Interner> Copy for PredicateKind<I>
 where
     I::DefId: Copy,
@@ -183,24 +182,6 @@ where
 {
 }
 
-impl<I: Interner> PartialEq for PredicateKind<I> {
-    fn eq(&self, other: &Self) -> bool {
-        match (self, other) {
-            (Self::Clause(l0), Self::Clause(r0)) => l0 == r0,
-            (Self::ObjectSafe(l0), Self::ObjectSafe(r0)) => l0 == r0,
-            (Self::Subtype(l0), Self::Subtype(r0)) => l0 == r0,
-            (Self::Coerce(l0), Self::Coerce(r0)) => l0 == r0,
-            (Self::ConstEquate(l0, l1), Self::ConstEquate(r0, r1)) => l0 == r0 && l1 == r1,
-            (Self::AliasRelate(l0, l1, l2), Self::AliasRelate(r0, r1, r2)) => {
-                l0 == r0 && l1 == r1 && l2 == r2
-            }
-            _ => core::mem::discriminant(self) == core::mem::discriminant(other),
-        }
-    }
-}
-
-impl<I: Interner> Eq for PredicateKind<I> {}
-
 impl<I: Interner> TypeFoldable<I> for PredicateKind<I>
 where
     I::DefId: TypeFoldable<I>,
diff --git a/tests/ui/traits/new-solver/alias-bound-unsound.rs b/tests/ui/traits/new-solver/alias-bound-unsound.rs
index 825e874d71b..907a3010355 100644
--- a/tests/ui/traits/new-solver/alias-bound-unsound.rs
+++ b/tests/ui/traits/new-solver/alias-bound-unsound.rs
@@ -23,7 +23,7 @@ fn main() {
     let x = String::from("hello, world");
     drop(<() as Foo>::copy_me(&x));
     //~^ ERROR overflow evaluating the requirement `<() as Foo>::Item: Sized`
-    //~| ERROR overflow evaluating the requirement `<() as Foo>::Item == _`
+    //~| ERROR overflow evaluating the requirement `<() as Foo>::Item normalizes-to _`
     //~| ERROR overflow evaluating the requirement `<() as Foo>::Item well-formed`
     //~| ERROR overflow evaluating the requirement `String <: <() as Foo>::Item`
     //~| ERROR overflow evaluating the requirement `&<() as Foo>::Item well-formed`
diff --git a/tests/ui/traits/new-solver/alias-bound-unsound.stderr b/tests/ui/traits/new-solver/alias-bound-unsound.stderr
index ca4b5c90ff2..29d4d983c03 100644
--- a/tests/ui/traits/new-solver/alias-bound-unsound.stderr
+++ b/tests/ui/traits/new-solver/alias-bound-unsound.stderr
@@ -19,7 +19,7 @@ LL |     drop(<() as Foo>::copy_me(&x));
    |
    = help: consider increasing the recursion limit by adding a `#![recursion_limit = "256"]` attribute to your crate (`alias_bound_unsound`)
 
-error[E0275]: overflow evaluating the requirement `<() as Foo>::Item == _`
+error[E0275]: overflow evaluating the requirement `<() as Foo>::Item normalizes-to _`
   --> $DIR/alias-bound-unsound.rs:24:10
    |
 LL |     drop(<() as Foo>::copy_me(&x));
diff --git a/tests/ui/traits/new-solver/generalize/generalize-proj-new-universe-index-2.rs b/tests/ui/traits/new-solver/generalize/generalize-proj-new-universe-index-2.rs
index 94d645a9859..7fe242f4714 100644
--- a/tests/ui/traits/new-solver/generalize/generalize-proj-new-universe-index-2.rs
+++ b/tests/ui/traits/new-solver/generalize/generalize-proj-new-universe-index-2.rs
@@ -1,5 +1,7 @@
 // compile-flags: -Ztrait-solver=next
 // known-bug: trait-system-refactor-initiative#60
+// dont-check-failure-status
+// dont-check-compiler-stderr
 
 // Generalizing a projection containing an inference variable
 // which cannot be named by the `root_vid` can result in ambiguity.
diff --git a/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.next.stderr b/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.next.stderr
index 34c2f0438c7..ad8b24a39c7 100644
--- a/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.next.stderr
+++ b/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.next.stderr
@@ -1,9 +1,11 @@
-error[E0271]: type mismatch resolving `<<T as Id<_>>::Id as Unnormalizable>::Assoc == _`
-  --> $DIR/occurs-check-nested-alias.rs:35:9
+error[E0275]: overflow evaluating the requirement `<<T as Id<_>>::Id as Unnormalizable>::Assoc == _`
+  --> $DIR/occurs-check-nested-alias.rs:36:9
    |
 LL |     x = y;
-   |         ^ types differ
+   |         ^
+   |
+   = help: consider increasing the recursion limit by adding a `#![recursion_limit = "256"]` attribute to your crate (`occurs_check_nested_alias`)
 
 error: aborting due to 1 previous error
 
-For more information about this error, try `rustc --explain E0271`.
+For more information about this error, try `rustc --explain E0275`.
diff --git a/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.rs b/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.rs
index a2113b2a8b3..02ac091c0a8 100644
--- a/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.rs
+++ b/tests/ui/traits/new-solver/generalize/occurs-check-nested-alias.rs
@@ -1,7 +1,8 @@
 // revisions: old next
 //[old] check-pass
 
-// Need to emit an alias-relate instead of a `Projection` goal here.
+// Currently always fails to generalize the outer alias, even if it
+// is treated as rigid by `alias-relate`.
 //[next] compile-flags: -Ztrait-solver=next
 //[next] known-bug: trait-system-refactor-initiative#8
 #![crate_type = "lib"]
diff --git a/tests/ui/traits/new-solver/specialization-transmute.stderr b/tests/ui/traits/new-solver/specialization-transmute.stderr
index 18965a465b3..eaf32a475ac 100644
--- a/tests/ui/traits/new-solver/specialization-transmute.stderr
+++ b/tests/ui/traits/new-solver/specialization-transmute.stderr
@@ -8,13 +8,11 @@ LL | #![feature(specialization)]
    = help: consider using `min_specialization` instead, which is more stable and complete
    = note: `#[warn(incomplete_features)]` on by default
 
-error[E0284]: type annotations needed
+error[E0284]: type annotations needed: cannot satisfy `<T as Default>::Id normalizes-to _`
   --> $DIR/specialization-transmute.rs:15:23
    |
 LL |     fn intu(&self) -> &Self::Id {
-   |                       ^^^^^^^^^ cannot infer type
-   |
-   = note: cannot satisfy `<T as Default>::Id == _`
+   |                       ^^^^^^^^^ cannot satisfy `<T as Default>::Id normalizes-to _`
 
 error[E0282]: type annotations needed
   --> $DIR/specialization-transmute.rs:13:23