Theorem rexbiia 2509

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 2505	1 |- ( E. x e. A ph <-> E. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   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:  2rexbiia  2510  ceqsrexbv  2891  reu8  2956  reldm  6239  djur  7128  prarloclem3  7557  suplocexprlemell  7773  recexgt0  8599  fsum3  11530  fprodseq  11726  even2n  12015  znf1o  14139  lmres  14416  reeff1o  14908  ioocosf1o  14989