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Theorem spcimgf 2874
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgf.1  |-  F/_ x A
spcimgf.2  |-  F/ x ps
spcimgf.3  |-  ( x  =  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spcimgf  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )

Proof of Theorem spcimgf
StepHypRef Expression
1 spcimgf.2 . . 3  |-  F/ x ps
2 spcimgf.1 . . 3  |-  F/_ x A
31, 2spcimgft 2872 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  V  ->  ( A. x ph  ->  ps ) ) )
4 spcimgf.3 . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
53, 4mpg 1538 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419
This theorem is referenced by:  spcimegf  2875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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