Theorem 2rexbiia 2549

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 2530	. 2 |- ( x e. A -> ( E. y e. B ph <-> E. y e. B ps ) )
3	2	rexbiia 2548	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 2202   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-rex 2517

This theorem is referenced by:  elq  9900  cnref1o  9929