Theorem 2rexbidva 2513

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 400	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps <-> ch ) )
3	2	rexbidva 2487	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps <-> E. y e. B ch ) )
4	3	rexbidva 2487	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 104   <-> wb 105   e. wcel 2160   E.wrex 2469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-rex 2474

This theorem is referenced by:  pythagtriplem2  12298  pythagtrip  12315