Theorem rexbiia 2565

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 458	. 2 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	rexbii2 2561	1 |- ( E. x e. A ph <-> E. x e. A ps )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   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:  2rexbiia  2566  ceqsrexbv  2957  reu8  3022  reldm  6420  djur  7410  prarloclem3  7865  suplocexprlemell  8081  recexgt0  8911  fsum3  12173  fprodseq  12369  even2n  12660  znf1o  15070  lmres  15440  reeff1o  15965  ioocosf1o  16047