Theorem 2rexbiia 1651

Description: Inference adding 2 restricted existential quantifiers to both sides of an equivalence.

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

Proof of Theorem 2rexbiia

Step	Hyp	Ref	Expression
1		2rexbiia.1	. . 3 |- ((x e. A /\ y e. B) -> (ph <-> ps))
2	1	rexbidva 1636	. 2 |- (x e. A -> (E.y e. B ph <-> E.y e. B ps))
3	2	rexbiia 1650	1 |- (E.x e. A E.y e. B ph <-> E.x e. A E.y e. B ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 1105  E.wrex 1622

This theorem is referenced by:  sqr2irr 6610  mdsymlem8 10459

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-gen 955  ax-17 1190

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-rex 1626

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