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Theorem rspcva 2814
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
Hypothesis
Ref Expression
rspcv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rspcva  |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rspcva
StepHypRef Expression
1 rspcv.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21rspcv 2812 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) )
32imp 123 1  |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   A.wral 2435
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714
This theorem is referenced by:  supmoti  6937  peano2nnnn  7773  squeeze0  8775  peano2nn  8845  nnsub  8872  zextle  9255  rexuz3  10890  cau3lem  11014  caubnd2  11017  climcn1  11205  dvdsext  11746
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