Theorem rexbii2 2561

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 1658	. 2 |- ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ps ) )
3		df-rex 2534	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4		df-rex 2534	. 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 )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-rex 2534

This theorem is used by:  rexeqbii  2563  rexbiia  2565  rexrab  2989  rexdifpr  3737  rexdifsn  3846  bnd2  4310  suplocsrlemb  8173  rexuz2  9981  rexrp  10077  rexuz3  11756  4sqexercise1  13177