Theorem 2rexbidva 2488

Description: Formula-building rule for restricted existential quantifiers (deduction form). (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 398	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps <-> ch ) )
3	2	rexbidva 2462	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps <-> E. y e. B ch ) )
4	3	rexbidva 2462	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 103   <-> wb 104   e. wcel 2136   E.wrex 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-rex 2449

This theorem is referenced by:  pythagtriplem2  12194  pythagtrip  12211