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Theorem spcv 3034
Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
Hypotheses
Ref Expression
spcv.1  |-  A  e. 
_V
spcv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcv  |-  ( A. x ph  ->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem spcv
StepHypRef Expression
1 spcv.1 . 2  |-  A  e. 
_V
2 spcv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32spcgv 3028 . 2  |-  ( A  e.  _V  ->  ( A. x ph  ->  ps ) )
41, 3ax-mp 8 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948
This theorem is referenced by:  morex  3110  rext  4404  relop  5015  frxp  6448  pssnn  7319  findcard  7339  fiint  7375  marypha1lem  7430  dfom3  7594  elom3  7595  aceq3lem  7993  dfac3  7994  dfac5lem4  7999  dfac8  8007  dfac9  8008  dfacacn  8013  dfac13  8014  kmlem1  8022  kmlem10  8031  fin23lem34  8218  fin23lem35  8219  zorn2lem7  8374  zornn0g  8377  axgroth6  8695  nnunb  10209  clatl  14535  gsumval3  15506  gsumzaddlem  15518  dfac14  17642  i1fd  19565  chlimi  22729  dfpo2  25370  dfon2lem4  25405  dfon2lem5  25406  dfon2lem7  25408  ttac  27098  dfac11  27128  dfac21  27132  symggen  27379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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