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Theorem vtoclgf 3134
Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgf.1  |-  F/_ x A
vtoclgf.2  |-  F/ x ps
vtoclgf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclgf.4  |-  ph
Assertion
Ref Expression
vtoclgf  |-  ( A  e.  V  ->  ps )

Proof of Theorem vtoclgf
StepHypRef Expression
1 elex 3087 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtoclgf.1 . . . 4  |-  F/_ x A
32issetf 3083 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 vtoclgf.2 . . . 4  |-  F/ x ps
5 vtoclgf.4 . . . . 5  |-  ph
6 vtoclgf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6mpbii 211 . . . 4  |-  ( x  =  A  ->  ps )
84, 7exlimi 1850 . . 3  |-  ( E. x  x  =  A  ->  ps )
93, 8sylbi 195 . 2  |-  ( A  e.  _V  ->  ps )
101, 9syl 16 1  |-  ( A  e.  V  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   E.wex 1587   F/wnf 1590    e. wcel 1758   F/_wnfc 2602   _Vcvv 3078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080
This theorem is referenced by:  vtocl2gf  3138  vtocl3gf  3139  vtoclgaf  3141  elabgf  3211  ssiun2sf  26081  subtr  28677  subtr2  28678  fmuldfeqlem1  29931  climsuse  29949  stoweidlem59  30022
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