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Theorem vtoclgf 2658
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 2611 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtoclgf.1 . . . 4  |-  F/_ x A
32issetf 2607 . . 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 1526 . . 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 1285   F/wnf 1390   E.wex 1422    e. wcel 1434   F/_wnfc 2207   _Vcvv 2602
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by:  vtoclg  2659  vtocl2gf  2661  vtocl3gf  2662  vtoclgaf  2664  ceqsexg  2724  elabgf  2737  mob  2775  opeliunxp2  4504  fvmptss2  5279
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