Theorem rexbii2 2518

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 1629	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ps ) )
3		df-rex 2491	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2491	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
5	2, 3, 4	3bitr4i 212	1 |- ( E. x e. A ph <-> E. x e. B ps )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1516   e. wcel 2177   E.wrex 2486

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-rex 2491

This theorem is referenced by:  rexeqbii  2520  rexbiia  2522  rexrab  2940  rexdifpr  3666  rexdifsn  3771  bnd2  4225  suplocsrlemb  7939  rexuz2  9722  rexrp  9818  rexuz3  11376  4sqexercise1  12796