Theorem rexbiia 2647

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 618	. 2 |- ((x e. A /\ ph) <-> (x e. A /\ ps))
3	2	rexbii2 2643	1 |- (E.x e. A ph <-> E.x e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  E.wrex 2615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-rex 2620

This theorem is referenced by:  2rexbiia  2648  ceqsrexbv  2973  reu8  3032  phialllem1  4616  finnc  6243  nchoicelem11  6299  nchoicelem16  6304