Theorem bi2rexa 1230

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

Hypothesis

Ref	Expression
bi2rexa.1	|- ((x e. A /\ y e. B) -> (ph <-> ps))

Assertion

Ref	Expression
bi2rexa	|- (E.x e. A E.y e. B ph <-> E.x e. A E.y e. B ps)

Distinct variable group(s):   x,y   y,A

Proof of Theorem bi2rexa

Step	Hyp	Ref	Expression
1		bi2rexa.1	. . 3 |- ((x e. A /\ y e. B) -> (ph <-> ps))
2	1	birexdva 1216	. 2 |- (x e. A -> (E.y e. B ph <-> E.y e. B ps))
3	2	birexa 1229	1 |- (E.x e. A E.y e. B ph <-> E.x e. A E.y e. B ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  elrnoprab 3054  sqr2irr 4782  mdsymlem8 5783

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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