Theorem rexbidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019) (Proof shortened by Wolf Lammen, 10-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	pm5.32da	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
3	2	rexbidv2	|- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )