Theorem rexbii2 2446

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 1584	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ps ) )
3		df-rex 2422	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2422	. 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 1468   e. wcel 1480   E.wrex 2417

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-rex 2422

This theorem is referenced by:  rexeqbii  2448  rexbiia  2450  rexrab  2847  rexdifsn  3655  bnd2  4097  suplocsrlemb  7614  rexuz2  9376  rexrp  9464  rexuz3  10762