Theorem rexbidv2

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999)

	Ref	Expression
Hypothesis	rexbidv2.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
Assertion	rexbidv2	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		rexbidv2.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
2	1	exbidv	|- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. B /\ ch ) ) )
3		df-rex	|- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
4		df-rex	|- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
5	2 3 4	3bitr4g	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )