Theorem 2rexbidva 2347

Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)

Hypothesis

Ref	Expression
2ralbidva.1	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
2rexbidva	|- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )

Distinct variable groups:   x, y, ph   y, A

Allowed substitution hints:   ps( x, y)   ch( x, y)   A( x)   B( x, y)

Proof of Theorem 2rexbidva

Step	Hyp	Ref	Expression
1		2ralbidva.1	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps <-> ch ) )
2	1	anassrs 380	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps <-> ch ) )
3	2	rexbidva 2323	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps <-> E. y e. B ch ) )
4	3	rexbidva 2323	1 |- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-rex 2312

This theorem is referenced by: (None)