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Theorem vtoclgf 2817
Description: Implicit substitution of a class for a set variable, with bound-variable hypotheses in place of distinct 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 2771 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtoclgf.1 . . . 4  |-  F/_ x A
32issetf 2768 . . 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 204 . . . 4  |-  ( x  =  A  ->  ps )
84, 7exlimi 1781 . . 3  |-  ( E. x  x  =  A  ->  ps )
93, 8sylbi 189 . 2  |-  ( A  e.  _V  ->  ps )
101, 9syl 17 1  |-  ( A  e.  V  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   E.wex 1537   F/wnf 1539    = wceq 1619    e. wcel 1621   F/_wnfc 2381   _Vcvv 2763
This theorem is referenced by:  vtoclg  2818  vtocl2gf  2820  vtocl3gf  2821  vtoclgaf  2823  ceqsexg  2874  elabgf  2887  mob  2922  opeliunxp2  4812  fvopab5  6255  dvfsumlem2  19336  dvfsumlem4  19338  subtr  25591  subtr2  25592  stoweidlem51  27135  stoweidlem59  27143
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765
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