Theorem 2rexbii 2479

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 2477	. 2 |- ( E. y e. B ph <-> E. y e. B ps )
3	2	rexbii 2477	1 |- ( E. x e. A E. y e. B ph <-> E. x e. A E. y e. B ps )

Syntax hints:   <-> wb 104   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-rex 2454

This theorem is referenced by:  3reeanv  2640  4fvwrd4  10096  prodmodc  11541  pythagtriplem2  12220  pythagtrip  12237