Theorem rexbida

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003)

	Ref	Expression
Hypotheses	rexbida.1	|- F/ x ph
	rexbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
Assertion	rexbida	|- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		rexbida.1	|- F/ x ph
2		rexbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.32da	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1 3	exbid	|- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ ch ) ) )
5		df-rex	|- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
6		df-rex	|- ( E. x e. A ch <-> E. x ( x e. A /\ ch ) )
7	4 5 6	3bitr4g	|- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )