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

Theorem spcegf 3024
Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
Hypotheses
Ref Expression
spcgf.1  |-  F/_ x A
spcgf.2  |-  F/ x ps
spcgf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcegf  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)

Proof of Theorem spcegf
StepHypRef Expression
1 spcgf.1 . . . 4  |-  F/_ x A
2 spcgf.2 . . . . 5  |-  F/ x ps
32nfn 1811 . . . 4  |-  F/ x  -.  ps
4 spcgf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54notbid 286 . . . 4  |-  ( x  =  A  ->  ( -.  ph  <->  -.  ps )
)
61, 3, 5spcgf 3023 . . 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    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558
This theorem is referenced by:  spcegv  3029  rspce  3039  euotd  4449  rspcegf  27651  stoweidlem36  27742  stoweidlem46  27752  bnj607  29214  bnj1491  29353
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