Theorem 2rexbii 2506

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 2504	. 2 |- ( E. y e. B ph <-> E. y e. B ps )
3	2	rexbii 2504	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 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-rex 2481

This theorem is referenced by:  3reeanv  2668  4fvwrd4  10215  prodmodc  11743  pythagtriplem2  12435  pythagtrip  12452