ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtocl2gf Unicode version

Theorem vtocl2gf 2799
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 2748 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtocl2gf.3 . . 3  |-  F/_ y B
3 vtocl2gf.2 . . . . 5  |-  F/_ y A
43nfel1 2330 . . . 4  |-  F/ y  A  e.  _V
5 vtocl2gf.5 . . . 4  |-  F/ y ch
64, 5nfim 1572 . . 3  |-  F/ y ( A  e.  _V  ->  ch )
7 vtocl2gf.7 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
87imbi2d 230 . . 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 2795 . . 3  |-  ( A  e.  _V  ->  ps )
142, 6, 8, 13vtoclgf 2795 . 2  |-  ( B  e.  W  ->  ( A  e.  _V  ->  ch ) )
151, 14mpan9 281 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   F/wnf 1460    e. wcel 2148   F/_wnfc 2306   _Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  vtocl3gf  2800  vtocl2g  2801  vtocl2gaf  2804
  Copyright terms: Public domain W3C validator