Theorem rexbii2 2390

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
rexbii2.1	|- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) )

Assertion

Ref	Expression
rexbii2	|- ( E. x e. A ph <-> E. x e. B ps )

Proof of Theorem rexbii2

Step	Hyp	Ref	Expression
1		rexbii2.1	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. B /\ ps ) )
2	1	exbii 1542	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ps ) )
3		df-rex 2366	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2366	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
5	2, 3, 4	3bitr4i 211	1 |- ( E. x e. A ph <-> E. x e. B ps )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1427   e. wcel 1439   E.wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-rex 2366

This theorem is referenced by:  rexeqbii  2392  rexbiia  2394  rexrab  2779  rexdifsn  3578  bnd2  4014  rexuz2  9123  rexrp  9210  rexuz3  10477