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Theorem vtoclgf 2780
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 2735 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtoclgf.1 . . . 4  |-  F/_ x A
32issetf 2732 . . 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 2372   _Vcvv 2727
This theorem is referenced by:  vtoclg  2781  vtocl2gf  2783  vtocl3gf  2784  vtoclgaf  2786  ceqsexg  2836  elabgf  2849  mob  2884  opeliunxp2  4731  fvopab5  6173  dvfsumlem2  19206  dvfsumlem4  19208  subtr  25390  subtr2  25391  stoweidlem51  26934  stoweidlem59  26942
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729
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