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

Theorem spcimegf 3022
Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgf.1  |-  F/_ x A
spcimgf.2  |-  F/ x ps
spcimegf.3  |-  ( x  =  A  ->  ( ps  ->  ph ) )
Assertion
Ref Expression
spcimegf  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)

Proof of Theorem spcimegf
StepHypRef Expression
1 spcimgf.1 . . . 4  |-  F/_ x A
2 spcimgf.2 . . . . 5  |-  F/ x ps
32nfn 1811 . . . 4  |-  F/ x  -.  ps
4 spcimegf.3 . . . . 5  |-  ( x  =  A  ->  ( ps  ->  ph ) )
54con3d 127 . . . 4  |-  ( x  =  A  ->  ( -.  ph  ->  -.  ps )
)
61, 3, 5spcimgf 3021 . . 3  |-  ( A  e.  V  ->  ( A. x  -.  ph  ->  -. 
ps ) )
76con2d 109 . 2  |-  ( A  e.  V  ->  ( ps  ->  -.  A. x  -.  ph ) )
8 df-ex 1551 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
97, 8syl6ibr 219 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
  Copyright terms: Public domain W3C validator