MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl2gf Structured version   Unicode version

Theorem vtocl2gf 3027
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
Hypotheses
Ref Expression
vtocl2gf.1  |-  F/_ x A
vtocl2gf.2  |-  F/_ y A
vtocl2gf.3  |-  F/_ y B
vtocl2gf.4  |-  F/ x ps
vtocl2gf.5  |-  F/ y ch
vtocl2gf.6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl2gf.7  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl2gf.8  |-  ph
Assertion
Ref Expression
vtocl2gf  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )

Proof of Theorem vtocl2gf
StepHypRef Expression
1 elex 2976 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtocl2gf.3 . . 3  |-  F/_ y B
3 vtocl2gf.2 . . . . 5  |-  F/_ y A
43nfel1 2584 . . . 4  |-  F/ y  A  e.  _V
5 vtocl2gf.5 . . . 4  |-  F/ y ch
64, 5nfim 1852 . . 3  |-  F/ y ( A  e.  _V  ->  ch )
7 vtocl2gf.7 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
87imbi2d 316 . . 3  |-  ( y  =  B  ->  (
( A  e.  _V  ->  ps )  <->  ( A  e.  _V  ->  ch )
) )
9 vtocl2gf.1 . . . 4  |-  F/_ x A
10 vtocl2gf.4 . . . 4  |-  F/ x ps
11 vtocl2gf.6 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
12 vtocl2gf.8 . . . 4  |-  ph
139, 10, 11, 12vtoclgf 3023 . . 3  |-  ( A  e.  _V  ->  ps )
142, 6, 8, 13vtoclgf 3023 . 2  |-  ( B  e.  W  ->  ( A  e.  _V  ->  ch ) )
151, 14mpan9 469 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2561   _Vcvv 2967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969
This theorem is referenced by:  vtocl3gf  3028  vtocl2g  3029  vtocl2gaf  3032  offval22  6647  fmuldfeqlem1  29716
  Copyright terms: Public domain W3C validator