Theorem rexeq

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025)

	Ref	Expression
Assertion	rexeq	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
2		anbi1	|- ( ( x e. A <-> x e. B ) -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
3	2	alexbii	|- ( A. x ( x e. A <-> x e. B ) -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
4	1 3	sylbi	|- ( A = B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
5		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		df-rex	|- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
7	4 5 6	3bitr4g	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )