Theorem rexbii2

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004)

	Ref	Expression
Hypothesis	rexbii2.1	|- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) )
Assertion	rexbii2	|- ( E. x e. A ph <-> E. x e. B ps )

Proof

Step	Hyp	Ref	Expression
1		rexbii2.1	|- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) )
2	1	exbii	|- ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ps ) )
3		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex	|- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
5	2 3 4	3bitr4i	|- ( E. x e. A ph <-> E. x e. B ps )