Theorem rexbii2 2505

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 1616	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ps ) )
3		df-rex 2478	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2478	. 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 1503   e. wcel 2164   E.wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-rex 2478

This theorem is referenced by:  rexeqbii  2507  rexbiia  2509  rexrab  2924  rexdifpr  3647  rexdifsn  3751  bnd2  4203  suplocsrlemb  7868  rexuz2  9649  rexrp  9745  rexuz3  11137  4sqexercise1  12539