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Theorem rspccv 2760
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 2759 . 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 1316    e. wcel 1465   A.wral 2393
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662
This theorem is referenced by:  elinti  3750  ofrval  5960  supubti  6854  suplubti  6855  suplocsrlempr  7583  pitonn  7624  peano5uzti  9127  zindd  9137  basis2  12142  tg2  12156  mopni  12578  metrest  12602  metcnpi  12611  metcnpi2  12612  decidi  12929  sumdc2  12933
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