diff --git a/theories/bi/updates.v b/theories/bi/updates.v
index d35e4007968df31d3e72fb7952e0d41de26e8099..e3496c94d92f795e9496b107dff28e0ef6f494a4 100644
--- a/theories/bi/updates.v
+++ b/theories/bi/updates.v
@@ -222,6 +222,25 @@ Section fupd_derived.
   Proof.
     intros (Ef&->&?)%subseteq_disjoint_union_L. by apply fupd_mask_frame_r.
   Qed.
+  (** How to apply an arbitrary mask-changing view shift when having
+      an arbitrary mask. *)
+  Lemma fupd_mask_frame E E1 E2 P :
+    E1 âŠ† E â†’ (|={E1,E2}=> P) -âˆ— (|={E,E2 âˆª (E âˆ– E1)}=> P).
+  Proof.
+    intros ?. rewrite (fupd_mask_frame_r _ _ (E âˆ– E1)); last set_solver.
+    assert (E = E1 âˆª E âˆ– E1) as <-; last done.
+    apply union_difference_L. done.
+  Qed.
+  Lemma fupd_mask_frame_diff E E1 E2 P :
+    (* E2 âŠ† E1 is needed bcause otherwise the [E âˆ– E2] could remove
+       more invariants from E than it did from E1. *)
+    E2 âŠ† E1 â†’ E1 âŠ† E â†’ (|={E1,E1âˆ–E2}=> P) -âˆ— (|={E,Eâˆ–E2}=> P).
+  Proof.
+    intros ? HE. rewrite (fupd_mask_frame E); last done.
+    assert (E1 âˆ– E2 âˆª E âˆ– E1 = E âˆ– E2) as <-; last done.
+    apply (anti_symm (âŠ†)); first set_solver.
+    rewrite {1}(union_difference_L _ _ HE). set_solver.
+  Qed.
 
   Lemma fupd_sep E P Q : (|={E}=> P) âˆ— (|={E}=> Q) ={E}=âˆ— P âˆ— Q.
   Proof. by rewrite fupd_frame_r fupd_frame_l fupd_trans. Qed.