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Theorem cla4gv 2805
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 2385 . 2  |-  F/_ x A
2 nfv 1629 . 2  |-  F/ x ps
3 cla4gv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3cla4gf 2801 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  2811  mob2  2882  intss1  3775  dfiin2g  3834  fri  4248  alxfr  4438  tfisi  4540  limomss  4552  nnlim  4560  isofrlem  5689  f1oweALT  5703  pssnn  6966  findcard3  6985  ttukeylem1  8020  rami  12936  ramcl  12950  clatlem  14060  islbs3  15742  mplsubglem  16011  mpllsslem  16012  uniopn  16475  chlimi  21644  relexpind  23208  dfon2lem3  23309  dfon2lem8  23314  sgplpte21b  25300  neificl  25633  ismrcd1  25939
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729
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