proposition: Lemma HF_eq_elem T x y x' y' : HFeq <<= T -> T ⊢ x ≡ x' -> T ⊢ y ≡ y' -> T ⊢ x ∈ y -> T ⊢ x' ∈ y'.
proof: Proof. intros H1 H2 H3 H4. eapply IE; try apply H4. eapply IE; try apply H3. eapply IE; try apply H2. assert (H : T ⊢ ax_eq_elem) by (apply Ctx; firstorder). now apply (AllE x), (AllE y), (AllE x'), (AllE y') in H; cbn in H; subsimpl_in H. Qed.
imports: Undecidability.FOL.ZF Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Sets.ZF Undecidability.FOL.Reductions.PCPb_to_HF Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation)