Theorem rexbiia 2339

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 427	. 2 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	rexbii2 2335	1 |- ( E. x e. A ph <-> E. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-rex 2312

This theorem is referenced by:  2rexbiia  2340  ceqsrexbv  2675  reu8  2737  reldm  5812  prarloclem3  6595  recexgt0  7571