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Theorem spcgv 3004
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 2548 . 2  |-  F/_ x A
2 nfv 1626 . 2  |-  F/ x ps
3 spcgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3spcgf 2999 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  3010  mob2  3082  intss1  4033  dfiin2g  4092  fri  4512  alxfr  4703  tfisi  4805  limomss  4817  nnlim  4825  isofrlem  6027  f1oweALT  6041  pssnn  7294  findcard3  7317  ttukeylem1  8353  rami  13346  ramcl  13360  clatlem  14502  islbs3  16190  mplsubglem  16461  mpllsslem  16462  uniopn  16933  chlimi  22698  relexpind  25101  dfon2lem3  25363  dfon2lem8  25368  neificl  26357  ismrcd1  26650
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 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926
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