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Theorem spcv 3115
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 3109 . 2  |-  ( A  e.  _V  ->  ( A. x ph  ->  ps ) )
41, 3ax-mp 5 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437    e. wcel 1872   _Vcvv 3022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-v 3024
This theorem is referenced by:  morex  3197  rext  4612  relop  4947  frxp  6861  pssnn  7743  findcard  7763  fiint  7801  marypha1lem  7900  dfom3  8105  elom3  8106  aceq3lem  8502  dfac3  8503  dfac5lem4  8508  dfac8  8516  dfac9  8517  dfacacn  8522  dfac13  8523  kmlem1  8531  kmlem10  8540  fin23lem34  8727  fin23lem35  8728  zorn2lem7  8883  zornn0g  8886  axgroth6  9204  nnunb  10816  symggen  17054  gsumval3lem2  17483  gsumzaddlem  17497  dfac14  20575  i1fd  22581  chlimi  26829  ddemeas  29011  dfpo2  30346  dfon2lem4  30383  dfon2lem5  30384  dfon2lem7  30386  ttac  35804  dfac11  35833  dfac21  35837
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