Theorem rexbiia 2485

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 451	. 2 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	rexbii2 2481	1 |- ( E. x e. A ph <-> E. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2141   E.wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by:  2rexbiia  2486  ceqsrexbv  2861  reu8  2926  reldm  6165  djur  7046  prarloclem3  7459  suplocexprlemell  7675  recexgt0  8499  fsum3  11350  fprodseq  11546  even2n  11833  lmres  13042  reeff1o  13488  ioocosf1o  13569