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Theorem vtoclgf 2677
Description: Implicit substitution of a class for a setvar 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 2630 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtoclgf.1 . . . 4  |-  F/_ x A
32issetf 2626 . . 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 146 . . . 4  |-  ( x  =  A  ->  ps )
84, 7exlimi 1530 . . 3  |-  ( E. x  x  =  A  ->  ps )
93, 8sylbi 119 . 2  |-  ( A  e.  _V  ->  ps )
101, 9syl 14 1  |-  ( A  e.  V  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   F/wnf 1394   E.wex 1426    e. wcel 1438   F/_wnfc 2215   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  vtoclg  2679  vtocl2gf  2681  vtocl3gf  2682  vtoclgaf  2684  ceqsexg  2743  elabgf  2756  mob  2795  opeliunxp2  4564  fvmptss2  5363  isummolem2a  10735
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