MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spcimgf Structured version   Unicode version

Theorem spcimgf 3029
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 3027 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  V  ->  ( A. x ph  ->  ps ) ) )
4 spcimgf.3 . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
53, 4mpg 1557 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2559
This theorem is referenced by:  spcimegf  3030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958
  Copyright terms: Public domain W3C validator