Theorem 2rexbiia 3055

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 3049	. 2 |- ( x e. A -> ( E. y e. B ph <-> E. y e. B ps ) )
3	2	rexbiia 3040	1 |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-rex 2918

This theorem is referenced by:  cnref1o  11827  mndpfo  17314  mdsymlem8  29269  xlt2addrd  29523  elunirnmbfm  30315  icoreelrnab  33202