proposition: Lemma prv_ex_eq { p : peirce } A x phi : binZF <<= A -> A ⊢ phi[x..] <-> A ⊢ ∃ $0 ≡' x`[↑] ∧ phi.
proof: Proof. intros HA. split; intros H. - apply (ExI x). unfold eq'. cbn. subsimpl. apply CI; trivial. now apply minZF_refl. - use_exists' H y. eapply minZF_Leibniz; eauto 3. apply minZF_sym; eauto 3. Qed.
imports: Undecidability.FOL.Syntax.Facts Undecidability.FOL.Syntax.Asimpl Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Deduction.FullNDFacts Undecidability.FOL.Sets(binZF ZF) Undecidability.FOL.Reductions.PCPb_to_ZF Undecidability(Shared.ListAutomation) Morphisms Undecidability.FOL(Reductions.PCPb_to_ZFeq)