Theorem rexeqf

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003) (Revised by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 9-Mar-2025)

	Ref	Expression
Hypotheses	raleqf.1	|- F/_ x A
	raleqf.2	|- F/_ x B
Assertion	rexeqf	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Proof

Step	Hyp	Ref	Expression
1		raleqf.1	|- F/_ x A
2		raleqf.2	|- F/_ x B
3	1 2	raleqf	|- ( A = B -> ( A. x e. A -. ph <-> A. x e. B -. ph ) )
4		ralnex	|- ( A. x e. A -. ph <-> -. E. x e. A ph )
5		ralnex	|- ( A. x e. B -. ph <-> -. E. x e. B ph )
6	3 4 5	3bitr3g	|- ( A = B -> ( -. E. x e. A ph <-> -. E. x e. B ph ) )
7	6	con4bid	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )