Theorem rexbiia 2472

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 450	. 2 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	rexbii2 2468	1 |- ( E. x e. A ph <-> E. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2128   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-rex 2441

This theorem is referenced by:  2rexbiia  2473  ceqsrexbv  2843  reu8  2908  reldm  6136  djur  7015  prarloclem3  7419  suplocexprlemell  7635  recexgt0  8459  fsum3  11295  fprodseq  11491  even2n  11777  lmres  12718  reeff1o  13164  ioocosf1o  13245