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Theorem spcimgf 2769
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 2765 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  V  ->  ( A. x ph  ->  ps ) ) )
4 spcimgf.3 . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
53, 4mpg 1428 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330    = wceq 1332   F/wnf 1437    e. wcel 1481   F/_wnfc 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691
This theorem is referenced by:  bj-nn0sucALT  13345
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