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Theorem spcgf 3152
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 3149 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  V  ->  ( A. x ph  ->  ps ) ) )
4 spcgf.3 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4mpg 1594 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-v 3074
This theorem is referenced by:  spcegf  3153  spcgv  3157  rspc  3167  elabgt  3204  eusvnf  4590  sumeq2w  13282  prodeq2w  27564
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