Theorem rexeqf

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003) (Revised by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Hypotheses	raleq1f.1	|- F/_ x A
	raleq1f.2	|- F/_ x B
Assertion	rexeqf	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Proof

Step	Hyp	Ref	Expression
1		raleq1f.1	|- F/_ x A
2		raleq1f.2	|- F/_ x B
3	1 2	nfeq	|- F/ x A = B
4		eleq2	|- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d	|- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3 5	exbid	|- ( A = B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
7		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex	|- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6 7 8	3bitr4g	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )