Theorem ralbiia 2452

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 179	. 2 |- ( ( x e. A -> ph ) <-> ( x e. A -> ps ) )
3	2	ralbii2 2448	1 |- ( A. x e. A ph <-> A. x e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1481   A.wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426

This theorem depends on definitions:  df-bi 116  df-ral 2422

This theorem is referenced by:  frind  4282  poinxp  4616  soinxp  4617  seinxp  4618  dffun8  5159  funcnv3  5193  fncnv  5197  fnres  5247  fvreseq  5532  isoini2  5728  smores  6197  resixp  6635  finomni  7020  caucvgre  10785