Theorem ralbiia 2647

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)

Hypothesis

Ref	Expression
ralbiia.1	|- (x e. A -> (ph <-> ps))

Assertion

Ref	Expression
ralbiia	|- (A.x e. A ph <-> A.x e. A ps)

Proof of Theorem ralbiia

Step	Hyp	Ref	Expression
1		ralbiia.1	. . 3 |- (x e. A -> (ph <-> ps))
2	1	pm5.74i 236	. 2 |- ((x e. A -> ph) <-> (x e. A -> ps))
3	2	ralbii2 2643	1 |- (A.x e. A ph <-> A.x e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  A.wral 2615

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

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  dffun8  5135  funcnv3  5158  fncnv  5159  fnres  5200  fvreseq  5399  isoini2  5499