Theorem ralbiia 2380

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 178	. 2 |- ( ( x e. A -> ph ) <-> ( x e. A -> ps ) )
3	2	ralbii2 2376	1 |- ( A. x e. A ph <-> A. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by:  frind  4107  poinxp  4427  soinxp  4428  seinxp  4429  dffun8  4949  funcnv3  4981  fncnv  4985  fnres  5035  fvreseq  5292  isoini2  5478  smores  5930  caucvgre  9867