MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtoclgf Unicode version

Theorem vtoclgf 2843
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 2797 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtoclgf.1 . . . 4  |-  F/_ x A
32issetf 2794 . . 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 1802 . . 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 1529   F/wnf 1532    = wceq 1624    e. wcel 1685   F/_wnfc 2407   _Vcvv 2789
This theorem is referenced by:  vtoclg  2844  vtocl2gf  2846  vtocl3gf  2847  vtoclgaf  2849  ceqsexg  2900  elabgf  2913  mob  2948  opeliunxp2  4823  fvopab5  6282  dvfsumlem2  19368  dvfsumlem4  19370  subtr  25623  subtr2  25624  stoweidlem51  27199  stoweidlem59  27207
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791
  Copyright terms: Public domain W3C validator