Theorem 2rexbii 2514

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)

Hypothesis

Ref	Expression
ralbii.1	|- ( ph <-> ps )

Assertion

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

Proof of Theorem 2rexbii

Step	Hyp	Ref	Expression
1		ralbii.1	. . 3 |- ( ph <-> ps )
2	1	rexbii 2512	. 2 |- ( E. y e. B ph <-> E. y e. B ps )
3	2	rexbii 2512	1 |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps )

Syntax hints:   <-> wb 105   E.wrex 2484

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-rex 2489

This theorem is referenced by:  3reeanv  2676  4fvwrd4  10261  prodmodc  11860  pythagtriplem2  12560  pythagtrip  12577