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

Theorem vtocl2gf 2674
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 2624 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 vtocl2gf.3 . . 3  |-  F/_ y B
3 vtocl2gf.2 . . . . 5  |-  F/_ y A
43nfel1 2235 . . . 4  |-  F/ y  A  e.  _V
5 vtocl2gf.5 . . . 4  |-  F/ y ch
64, 5nfim 1507 . . 3  |-  F/ y ( A  e.  _V  ->  ch )
7 vtocl2gf.7 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
87imbi2d 228 . . 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 2671 . . 3  |-  ( A  e.  _V  ->  ps )
142, 6, 8, 13vtoclgf 2671 . 2  |-  ( B  e.  W  ->  ( A  e.  _V  ->  ch ) )
151, 14mpan9 275 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287   F/wnf 1392    e. wcel 1436   F/_wnfc 2212   _Vcvv 2615
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  vtocl3gf  2675  vtocl2g  2676  vtocl2gaf  2679
  Copyright terms: Public domain W3C validator