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Theorem spcgv 2996
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
Hypothesis
Ref Expression
spcgv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcgv  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem spcgv
StepHypRef Expression
1 nfcv 2540 . 2  |-  F/_ x A
2 nfv 1626 . 2  |-  F/ x ps
3 spcgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3spcgf 2991 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1721
This theorem is referenced by:  spcv  3002  mob2  3074  intss1  4025  dfiin2g  4084  fri  4504  alxfr  4695  tfisi  4797  limomss  4809  nnlim  4817  isofrlem  6019  f1oweALT  6033  pssnn  7286  findcard3  7309  ttukeylem1  8345  rami  13338  ramcl  13352  clatlem  14494  islbs3  16182  mplsubglem  16453  mpllsslem  16454  uniopn  16925  chlimi  22690  relexpind  25093  dfon2lem3  25355  dfon2lem8  25360  neificl  26349  ismrcd1  26642
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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