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Theorem ralbiia 2381
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
StepHypRef Expression
1 ralbiia.1 . . 3  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.74i 178 . 2  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  ->  ps )
)
32ralbii2 2377 1  |-  ( A. x  e.  A  ph  <->  A. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1434   A.wral 2349
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 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115  df-ral 2354
This theorem is referenced by:  frind  4115  poinxp  4435  soinxp  4436  seinxp  4437  dffun8  4959  funcnv3  4992  fncnv  4996  fnres  5046  fvreseq  5303  isoini2  5489  smores  5941  caucvgre  10005
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