Theorem rexbiia 2512

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)

Hypothesis

Ref	Expression
ralbiia.1	|- ( x e. A -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem rexbiia

Step	Hyp	Ref	Expression
1		ralbiia.1	. . 3 |- ( x e. A -> ( ph <-> ps ) )
2	1	pm5.32i 454	. 2 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	rexbii2 2508	1 |- ( E. x e. A ph <-> E. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2167   E.wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by:  2rexbiia  2513  ceqsrexbv  2895  reu8  2960  reldm  6253  djur  7144  prarloclem3  7581  suplocexprlemell  7797  recexgt0  8624  fsum3  11569  fprodseq  11765  even2n  12056  znf1o  14283  lmres  14568  reeff1o  15093  ioocosf1o  15174