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Theorem cla4gv 2819
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
Hypothesis
Ref Expression
cla4gv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cla4gv  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem cla4gv
StepHypRef Expression
1 nfcv 2392 . 2  |-  F/_ x A
2 nfv 1629 . 2  |-  F/ x ps
3 cla4gv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3cla4gf 2814 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532    = wceq 1619    e. wcel 1621
This theorem is referenced by:  cla4v  2825  mob2  2896  intss1  3818  dfiin2g  3877  fri  4292  alxfr  4484  tfisi  4586  limomss  4598  nnlim  4606  isofrlem  5736  f1oweALT  5750  pssnn  7014  findcard3  7033  ttukeylem1  8069  rami  12989  ramcl  13003  clatlem  14143  islbs3  15835  mplsubglem  16106  mpllsslem  16107  uniopn  16570  chlimi  21739  relexpind  23374  dfon2lem3  23475  dfon2lem8  23480  sgplpte21b  25466  neificl  25799  ismrcd1  26105
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742
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