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Theorem spcv 3002
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 2996 . 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 1546    = wceq 1649    e. wcel 1721   _Vcvv 2916
This theorem is referenced by:  morex  3078  rext  4372  relop  4982  frxp  6415  pssnn  7286  findcard  7306  fiint  7342  marypha1lem  7396  dfom3  7558  elom3  7559  aceq3lem  7957  dfac3  7958  dfac5lem4  7963  dfac8  7971  dfac9  7972  dfacacn  7977  dfac13  7978  kmlem1  7986  kmlem10  7995  fin23lem34  8182  fin23lem35  8183  zorn2lem7  8338  zornn0g  8341  axgroth6  8659  nnunb  10173  clatl  14498  gsumval3  15469  gsumzaddlem  15481  dfac14  17603  i1fd  19526  chlimi  22690  dfpo2  25326  dfon2lem4  25356  dfon2lem5  25357  dfon2lem7  25359  ttac  26997  dfac11  27028  dfac21  27032  symggen  27279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918
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