diff --git a/base_logic/base_logic.v b/base_logic/base_logic.v
index 31e643b158b2007ef8fe1d75824ef7341660d365..2ea42a57a9fb5f9a38061ede335f4586cc885890 100644
--- a/base_logic/base_logic.v
+++ b/base_logic/base_logic.v
@@ -14,4 +14,4 @@ Hint Resolve and_intro and_elim_l' and_elim_r' : I.
 Hint Resolve always_mono : I.
 Hint Resolve sep_elim_l' sep_elim_r' sep_mono : I.
 Hint Immediate True_intro False_elim : I.
-Hint Immediate iff_refl eq_refl' : I.
+Hint Immediate iff_refl internal_eq_refl' : I.
diff --git a/base_logic/derived.v b/base_logic/derived.v
index 7f4efa6e25a760fc172de6ba5a00cb25374cf4d6..a485b335a8fc86af8a242fa4238b04b549e0be62 100644
--- a/base_logic/derived.v
+++ b/base_logic/derived.v
@@ -241,13 +241,13 @@ Proof.
   - apply exist_elim=> x. eauto using pure_mono.
 Qed.
 
-Lemma eq_refl' {A : cofeT} (a : A) P : P ⊢ a ≡ a.
-Proof. rewrite (True_intro P). apply eq_refl. Qed.
-Hint Resolve eq_refl'.
-Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊢ a ≡ b.
+Lemma internal_eq_refl' {A : cofeT} (a : A) P : P ⊢ a ≡ a.
+Proof. rewrite (True_intro P). apply internal_eq_refl. Qed.
+Hint Resolve internal_eq_refl'.
+Lemma equiv_internal_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊢ a ≡ b.
 Proof. by intros ->. Qed.
-Lemma eq_sym {A : cofeT} (a b : A) : a ≡ b ⊢ b ≡ a.
-Proof. apply (eq_rewrite a b (λ b, b ≡ a)%I); auto. solve_proper. Qed.
+Lemma internal_eq_sym {A : cofeT} (a b : A) : a ≡ b ⊢ b ≡ a.
+Proof. apply (internal_eq_rewrite a b (λ b, b ≡ a)%I); auto. solve_proper. Qed.
 
 Lemma pure_alt φ : ■ φ ⊣⊢ ∃ _ : φ, True.
 Proof.
@@ -280,9 +280,10 @@ Proof.
 Qed.
 Lemma equiv_iff P Q : (P ⊣⊢ Q) → True ⊢ P ↔ Q.
 Proof. intros ->; apply iff_refl. Qed.
-Lemma eq_iff P Q : P ≡ Q ⊢ P ↔ Q.
+Lemma internal_eq_iff P Q : P ≡ Q ⊢ P ↔ Q.
 Proof.
-  apply (eq_rewrite P Q (λ Q, P ↔ Q))%I; first solve_proper; auto using iff_refl.
+  apply (internal_eq_rewrite P Q (λ Q, P ↔ Q))%I;
+    first solve_proper; auto using iff_refl.
 Qed.
 
 (* Derived BI Stuff *)
@@ -445,12 +446,12 @@ Proof.
   apply impl_intro_l; rewrite -always_and.
   apply always_mono, impl_elim with P; auto.
 Qed.
-Lemma always_eq {A:cofeT} (a b : A) : □ (a ≡ b) ⊣⊢ a ≡ b.
+Lemma always_internal_eq {A:cofeT} (a b : A) : □ (a ≡ b) ⊣⊢ a ≡ b.
 Proof.
   apply (anti_symm (⊢)); auto using always_elim.
-  apply (eq_rewrite a b (λ b, □ (a ≡ b))%I); auto.
+  apply (internal_eq_rewrite a b (λ b, □ (a ≡ b))%I); auto.
   { intros n; solve_proper. }
-  rewrite -(eq_refl a) always_pure; auto.
+  rewrite -(internal_eq_refl a) always_pure; auto.
 Qed.
 
 Lemma always_and_sep P Q : □ (P ∧ Q) ⊣⊢ □ (P ★ Q).
@@ -692,8 +693,8 @@ Global Instance ownM_timeless (a : M) : Timeless a → TimelessP (uPred_ownM a).
 Proof.
   intros ?. rewrite /TimelessP later_ownM. apply exist_elim=> b.
   rewrite (timelessP (a≡b)) (except_0_intro (uPred_ownM b)) -except_0_and.
-  apply except_0_mono. rewrite eq_sym.
-  apply (eq_rewrite b a (uPred_ownM)); first apply _; auto.
+  apply except_0_mono. rewrite internal_eq_sym.
+  apply (internal_eq_rewrite b a (uPred_ownM)); first apply _; auto.
 Qed.
 
 (* Persistence *)
@@ -716,9 +717,9 @@ Proof. by intros; rewrite /PersistentP always_forall; apply forall_mono. Qed.
 Global Instance exist_persistent {A} (Ψ : A → uPred M) :
   (∀ x, PersistentP (Ψ x)) → PersistentP (∃ x, Ψ x).
 Proof. by intros; rewrite /PersistentP always_exist; apply exist_mono. Qed.
-Global Instance eq_persistent {A : cofeT} (a b : A) :
+Global Instance internal_eq_persistent {A : cofeT} (a b : A) :
   PersistentP (a ≡ b : uPred M)%I.
-Proof. by intros; rewrite /PersistentP always_eq. Qed.
+Proof. by intros; rewrite /PersistentP always_internal_eq. Qed.
 Global Instance cmra_valid_persistent {A : cmraT} (a : A) :
   PersistentP (✓ a : uPred M)%I.
 Proof. by intros; rewrite /PersistentP always_cmra_valid. Qed.
diff --git a/base_logic/primitive.v b/base_logic/primitive.v
index 5c8f15f1a70e24d36c2d0156a5310c68b6f85b93..7bcce8d77346757af140bb40b9f79321fcf881c8 100644
--- a/base_logic/primitive.v
+++ b/base_logic/primitive.v
@@ -58,12 +58,13 @@ Definition uPred_exist_aux : { x | x = @uPred_exist_def }. by eexists. Qed.
 Definition uPred_exist {M A} := proj1_sig uPred_exist_aux M A.
 Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := proj2_sig uPred_exist_aux.
 
-Program Definition uPred_eq_def {M} {A : cofeT} (a1 a2 : A) : uPred M :=
+Program Definition uPred_internal_eq_def {M} {A : cofeT} (a1 a2 : A) : uPred M :=
   {| uPred_holds n x := a1 ≡{n}≡ a2 |}.
 Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
-Definition uPred_eq_aux : { x | x = @uPred_eq_def }. by eexists. Qed.
-Definition uPred_eq {M A} := proj1_sig uPred_eq_aux M A.
-Definition uPred_eq_eq: @uPred_eq = @uPred_eq_def := proj2_sig uPred_eq_aux.
+Definition uPred_internal_eq_aux : { x | x = @uPred_internal_eq_def }. by eexists. Qed.
+Definition uPred_internal_eq {M A} := proj1_sig uPred_internal_eq_aux M A.
+Definition uPred_internal_eq_eq:
+  @uPred_internal_eq = @uPred_internal_eq_def := proj2_sig uPred_internal_eq_aux.
 
 Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}.
@@ -177,7 +178,7 @@ Notation "â–¡ P" := (uPred_always P)
   (at level 20, right associativity) : uPred_scope.
 Notation "â–· P" := (uPred_later P)
   (at level 20, right associativity) : uPred_scope.
-Infix "≡" := uPred_eq : uPred_scope.
+Infix "≡" := uPred_internal_eq : uPred_scope.
 Notation "✓ x" := (uPred_cmra_valid x) (at level 20) : uPred_scope.
 Notation "|==> Q" := (uPred_bupd Q)
   (at level 99, Q at level 200, format "|==>  Q") : uPred_scope.
@@ -189,7 +190,7 @@ Notation "P ==★ Q" := (P -★ |==> Q)%I
 Module uPred_primitive.
 Definition unseal :=
   (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
-  uPred_exist_eq, uPred_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq,
+  uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq,
   uPred_later_eq, uPred_ownM_eq, uPred_cmra_valid_eq, uPred_bupd_eq).
 Ltac unseal := rewrite !unseal /=.
 
@@ -245,15 +246,15 @@ Proof.
 Qed.
 Global Instance wand_proper :
   Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_wand M) := ne_proper_2 _.
-Global Instance eq_ne (A : cofeT) n :
-  Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
+Global Instance internal_eq_ne (A : cofeT) n :
+  Proper (dist n ==> dist n ==> dist n) (@uPred_internal_eq M A).
 Proof.
   intros x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *.
   - by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto.
   - by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto.
 Qed.
-Global Instance eq_proper (A : cofeT) :
-  Proper ((≡) ==> (≡) ==> (⊣⊢)) (@uPred_eq M A) := ne_proper_2 _.
+Global Instance internal_eq_proper (A : cofeT) :
+  Proper ((≡) ==> (≡) ==> (⊣⊢)) (@uPred_internal_eq M A) := ne_proper_2 _.
 Global Instance forall_ne A n :
   Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
 Proof.
@@ -355,21 +356,16 @@ Proof. unseal; split=> n x ??; by exists a. Qed.
 Lemma exist_elim {A} (Φ : A → uPred M) Q : (∀ a, Φ a ⊢ Q) → (∃ a, Φ a) ⊢ Q.
 Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed.
 
-Lemma eq_refl {A : cofeT} (a : A) : True ⊢ a ≡ a.
+Lemma internal_eq_refl {A : cofeT} (a : A) : True ⊢ a ≡ a.
 Proof. unseal; by split=> n x ??; simpl. Qed.
-Lemma eq_rewrite {A : cofeT} a b (Ψ : A → uPred M) P
+Lemma internal_eq_rewrite {A : cofeT} a b (Ψ : A → uPred M) P
   {HΨ : ∀ n, Proper (dist n ==> dist n) Ψ} : (P ⊢ a ≡ b) → (P ⊢ Ψ a) → P ⊢ Ψ b.
 Proof.
   unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto.
   - by symmetry; apply Hab with x.
   - by apply Ha.
 Qed.
-Lemma eq_equiv {A : cofeT} (a b : A) : (True ⊢ a ≡ b) → a ≡ b.
-Proof.
-  unseal=> Hab; apply equiv_dist; intros n; apply Hab with ∅; last done.
-  apply cmra_valid_validN, ucmra_unit_valid.
-Qed.
-Lemma eq_rewrite_contractive {A : cofeT} a b (Ψ : A → uPred M) P
+Lemma internal_eq_rewrite_contractive {A : cofeT} a b (Ψ : A → uPred M) P
   {HΨ : Contractive Ψ} : (P ⊢ ▷ (a ≡ b)) → (P ⊢ Ψ a) → P ⊢ Ψ b.
 Proof.
   unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto.
diff --git a/program_logic/boxes.v b/program_logic/boxes.v
index e2abbd055e972bc9eac6e223cee3db8e47acdaa8..03db8da2a658997873cdabedc3d0d495db509d17 100644
--- a/program_logic/boxes.v
+++ b/program_logic/boxes.v
@@ -165,7 +165,8 @@ Lemma box_fill_all f P Q : box N f P ★ ▷ P ={N}=★ box N (const true <$> f)
 Proof.
   iIntros "[H HP]"; iDestruct "H" as (Φ) "[#HeqP Hf]".
   iExists Φ; iSplitR; first by rewrite big_sepM_fmap.
-  rewrite eq_iff later_iff big_sepM_later; iDestruct ("HeqP" with "HP") as "HP".
+  rewrite internal_eq_iff later_iff big_sepM_later.
+  iDestruct ("HeqP" with "HP") as "HP".
   iCombine "Hf" "HP" as "Hf".
   rewrite big_sepM_fmap; iApply (fupd_big_sepM _ _ f).
   iApply (big_sepM_impl _ _ f); iFrame "Hf".
@@ -192,7 +193,7 @@ Proof.
     iMod ("Hclose" with "[Hγ]"); first (iNext; iExists false; iFrame; eauto).
     by iApply "HΦ". }
   iModIntro; iSplitL "HΦ".
-  - rewrite eq_iff later_iff big_sepM_later. by iApply "HeqP".
+  - rewrite internal_eq_iff later_iff big_sepM_later. by iApply "HeqP".
   - iExists Φ; iSplit; by rewrite big_sepM_fmap.
 Qed.
 End box.
diff --git a/program_logic/iprop.v b/program_logic/iprop.v
index 62c7ceb0586ba274a53641efd0ef6c0b626c9c77..96b7ea91164df204b244040023f4037e12f5bcc5 100644
--- a/program_logic/iprop.v
+++ b/program_logic/iprop.v
@@ -151,6 +151,6 @@ Lemma iProp_unfold_equivI {Σ} (P Q : iProp Σ) :
   iProp_unfold P ≡ iProp_unfold Q ⊢ (P ≡ Q : iProp Σ).
 Proof.
   rewrite -{2}(iProp_fold_unfold P) -{2}(iProp_fold_unfold Q).
-  eapply (uPred.eq_rewrite _ _ (λ z,
+  eapply (uPred.internal_eq_rewrite _ _ (λ z,
     iProp_fold (iProp_unfold P) ≡ iProp_fold z))%I; auto with I; solve_proper.
 Qed.
diff --git a/program_logic/saved_prop.v b/program_logic/saved_prop.v
index 5a804a2a1f6f4e72c77576db631043ac921308a8..22cfc001432763165dd80748d444718a09e4dc2c 100644
--- a/program_logic/saved_prop.v
+++ b/program_logic/saved_prop.v
@@ -42,7 +42,7 @@ Section saved_prop.
     { intros z. rewrite /G1 /G2 -cFunctor_compose -{2}[z]cFunctor_id.
       apply (ne_proper (cFunctor_map F)); split=>?; apply iProp_fold_unfold. }
     rewrite -{2}[x]help -{2}[y]help. apply later_mono.
-    apply (eq_rewrite (G1 x) (G1 y) (λ z, G2 (G1 x) ≡ G2 z))%I;
+    apply (internal_eq_rewrite (G1 x) (G1 y) (λ z, G2 (G1 x) ≡ G2 z))%I;
       first solve_proper; auto with I.
   Qed.
 End saved_prop.
diff --git a/proofmode/class_instances.v b/proofmode/class_instances.v
index 06cbcc6d01319ddbd86b9428eea5089f9889fb7c..f03509b43dfe354077708e44c46fa00b66982042 100644
--- a/proofmode/class_instances.v
+++ b/proofmode/class_instances.v
@@ -33,8 +33,11 @@ Proof. by rewrite /IntoPure discrete_valid. Qed.
 (* FromPure *)
 Global Instance from_pure_pure φ : @FromPure M (■ φ) φ.
 Proof. done. Qed.
-Global Instance from_pure_eq {A : cofeT} (a b : A) : @FromPure M (a ≡ b) (a ≡ b).
-Proof. rewrite /FromPure. eapply pure_elim; [done|]=> ->. apply eq_refl'. Qed.
+Global Instance from_pure_internal_eq {A : cofeT} (a b : A) :
+  @FromPure M (a ≡ b) (a ≡ b).
+Proof.
+  rewrite /FromPure. eapply pure_elim; [done|]=> ->. apply internal_eq_refl'.
+Qed.
 Global Instance from_pure_cmra_valid {A : cmraT} (a : A) :
   @FromPure M (✓ a) (✓ a).
 Proof.
diff --git a/proofmode/coq_tactics.v b/proofmode/coq_tactics.v
index f55b0e46e997ae55f057540078c33ed6261b2f7a..d520cf9a830bb706ac991aac02b0e4ff0adfc528 100644
--- a/proofmode/coq_tactics.v
+++ b/proofmode/coq_tactics.v
@@ -653,9 +653,9 @@ Lemma tac_rewrite Δ i p Pxy (lr : bool) Q :
     (∀ n, Proper (dist n ==> dist n) Φ) →
     (Δ ⊢ Φ (if lr then x else y)) → Δ ⊢ Q.
 Proof.
-  intros ? A x y ? HPxy -> ?; apply eq_rewrite; auto.
+  intros ? A x y ? HPxy -> ?; apply internal_eq_rewrite; auto.
   rewrite {1}envs_lookup_sound' //; rewrite sep_elim_l HPxy.
-  destruct lr; auto using eq_sym.
+  destruct lr; auto using internal_eq_sym.
 Qed.
 
 Lemma tac_rewrite_in Δ i p Pxy j q P (lr : bool) Q :
@@ -673,8 +673,10 @@ Proof.
   rewrite sep_elim_l HPxy always_and_sep_r.
   rewrite (envs_simple_replace_sound _ _ j) //; simpl.
   rewrite HP right_id -assoc; apply wand_elim_r'. destruct lr.
-  - apply (eq_rewrite x y (λ y, □?q Φ y -★ Δ')%I); eauto with I. solve_proper.
-  - apply (eq_rewrite y x (λ y, □?q Φ y -★ Δ')%I); eauto using eq_sym with I.
+  - apply (internal_eq_rewrite x y (λ y, □?q Φ y -★ Δ')%I);
+      eauto with I. solve_proper.
+  - apply (internal_eq_rewrite y x (λ y, □?q Φ y -★ Δ')%I);
+      eauto using internal_eq_sym with I.
     solve_proper.
 Qed.
 
diff --git a/proofmode/tactics.v b/proofmode/tactics.v
index 1f449b36ab698c6e6ff65dde25d9188ec635e860..23848b1b8fb4c313707da9ddb3693049cc797ad0 100644
--- a/proofmode/tactics.v
+++ b/proofmode/tactics.v
@@ -1200,7 +1200,7 @@ Tactic Notation "iMod" open_constr(lem) "as" "%" simple_intropattern(pat) :=
 Hint Extern 0 (of_envs _ ⊢ _) => by iPureIntro.
 Hint Extern 0 (of_envs _ ⊢ _) => iAssumption.
 Hint Extern 0 (of_envs _ ⊢ _) => progress iIntros.
-Hint Resolve uPred.eq_refl'. (* Maybe make an [iReflexivity] tactic *)
+Hint Resolve uPred.internal_eq_refl'. (* Maybe make an [iReflexivity] tactic *)
 
 (* We should be able to write [Hint Extern 1 (of_envs _ ⊢ (_ ★ _)%I) => ...],
 but then [eauto] mysteriously fails. See bug 4762 *)
diff --git a/tests/proofmode.v b/tests/proofmode.v
index f3c087059015d0818d47b17f80c6c5b05d28e114..7466cef64c69b80d2876b5500d58224d0db54815 100644
--- a/tests/proofmode.v
+++ b/tests/proofmode.v
@@ -67,7 +67,7 @@ Lemma demo_5 (M : ucmraT) (x y : M) (P : uPred M) :
   (∀ z, P → z ≡ y) ⊢ (P -★ (x,x) ≡ (y,x)).
 Proof.
   iIntros "H1 H2".
-  iRewrite (uPred.eq_sym x x with "[#]"); first done.
+  iRewrite (uPred.internal_eq_sym x x with "[#]"); first done.
   iRewrite -("H1" $! _ with "[-]"); first done.
   done.
 Qed.