Theorem 2rexbiia 2510

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x, y   y, A

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

Proof of Theorem 2rexbiia

Step	Hyp	Ref	Expression
1		2rexbiia.1	. . 3 |- ( ( x e. A /\ y e. B ) -> ( ph <-> ps ) )
2	1	rexbidva 2491	. 2 |- ( x e. A -> ( E. y e. B ph <-> E. y e. B ps ) )
3	2	rexbiia 2509	1 |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   E.wrex 2473

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 2478

This theorem is referenced by:  elq  9687  cnref1o  9716