Theorem ralbiia 2338

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 169	. 2 |- ( ( x e. A -> ph ) <-> ( x e. A -> ps ) )
3	2	ralbii2 2334	1 |- ( A. x e. A ph <-> A. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1393   A.wral 2306

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

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  frind  4089  poinxp  4409  soinxp  4410  seinxp  4411  dffun8  4929  funcnv3  4961  fncnv  4965  fnres  5015  fvreseq  5271  isoini2  5458  smores  5907  caucvgre  9580