Theorem raleq

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-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	raleq	|- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )

Proof

Step	Hyp	Ref	Expression
1		rexeq	|- ( A = B -> ( E. x e. A -. ph <-> E. x e. B -. ph ) )
2		rexnal	|- ( E. x e. A -. ph <-> -. A. x e. A ph )
3		rexnal	|- ( E. x e. B -. ph <-> -. A. x e. B ph )
4	1 2 3	3bitr3g	|- ( A = B -> ( -. A. x e. A ph <-> -. A. x e. B ph ) )
5	4	con4bid	|- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )