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Theorem rspccv 2755
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
Hypothesis
Ref Expression
rspcv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rspccv  |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rspccv
StepHypRef Expression
1 rspcv.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21rspcv 2754 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) )
32com12 30 1  |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1312    e. wcel 1461   A.wral 2388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657
This theorem is referenced by:  elinti  3744  ofrval  5944  supubti  6836  suplubti  6837  pitonn  7577  peano5uzti  9057  zindd  9067  basis2  12052  tg2  12066  mopni  12465  metrest  12489  metcnpi  12498  metcnpi2  12499  decidi  12685  sumdc2  12689
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