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Theorem cla4gf 2814
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
Hypotheses
Ref Expression
cla4gf.1  |-  F/_ x A
cla4gf.2  |-  F/ x ps
cla4gf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cla4gf  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )

Proof of Theorem cla4gf
StepHypRef Expression
1 cla4gf.2 . . 3  |-  F/ x ps
2 cla4gf.1 . . 3  |-  F/_ x A
31, 2cla4gft 2811 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  V  ->  ( A. x ph  ->  ps ) ) )
4 cla4gf.3 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4mpg 1542 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532   F/wnf 1539    = wceq 1619    e. wcel 1621   F/_wnfc 2379
This theorem is referenced by:  cla4egf  2815  cla4gv  2819  rcla4  2829  elabgt  2862  eusvnf  4466  sumeq2w  12095
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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