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Theorem spcgf 3032
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
spcgf.1  |-  F/_ x A
spcgf.2  |-  F/ x ps
spcgf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcgf  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )

Proof of Theorem spcgf
StepHypRef Expression
1 spcgf.2 . . 3  |-  F/ x ps
2 spcgf.1 . . 3  |-  F/_ x A
31, 2spcgft 3029 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  V  ->  ( A. x ph  ->  ps ) ) )
4 spcgf.3 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4mpg 1558 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   F/wnf 1554    = wceq 1653    e. wcel 1726   F/_wnfc 2560
This theorem is referenced by:  spcegf  3033  spcgv  3037  rspc  3047  elabgt  3080  eusvnf  4719  sumeq2w  12487  prodeq2w  25239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959
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