reserve Y for non empty set;

theorem Th21: :: Frege's law
  for a,b,c being Function of Y,BOOLEAN holds (a 'imp' (b
  'imp' c)) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
    let x be Element of Y;
A1: now
      per cases by XBOOLEAN:def 3;
      case
        a.x=TRUE;
        hence ('not' a.x 'or' a.x)=TRUE by BINARITH:10;
      end;
      case
        a.x=FALSE;
        then 'not' a.x 'or' a.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' a.x 'or' a.x)=TRUE;
      end;
    end;
A2: now
      per cases by XBOOLEAN:def 3;
      case
        c.x=TRUE;
        hence ('not' c.x 'or' c.x)=TRUE by BINARITH:10;
      end;
      case
        c.x=FALSE;
        then 'not' c.x 'or' c.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' c.x 'or' c.x)=TRUE;
      end;
    end;
A3: now
      per cases by XBOOLEAN:def 3;
      case
        b.x=TRUE;
        hence ('not' b.x 'or' b.x)=TRUE by BINARITH:10;
      end;
      case
        b.x=FALSE;
        then 'not' b.x 'or' b.x =TRUE 'or' FALSE by MARGREL1:11
          .=TRUE by BINARITH:10;
        hence ('not' b.x 'or' b.x)=TRUE;
      end;
    end;
    ((a 'imp' (b 'imp' c)) 'imp' ((a 'imp' b) 'imp' (a 'imp' c))).x =
    'not' (a 'imp' (b 'imp' c)).x 'or' ((a 'imp' b) 'imp' (a 'imp' c)).x by
BVFUNC_1:def 8
      .='not' ('not' a.x 'or' (b 'imp' c).x) 'or' ((a 'imp' b) 'imp' (a
    'imp' c)).x by BVFUNC_1:def 8
      .='not' ('not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ((a 'imp' b) 'imp'
    (a 'imp' c)).x by BVFUNC_1:def 8
      .='not' ('not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ('not' (a 'imp' b).
    x 'or' (a 'imp' c).x) by BVFUNC_1:def 8
      .='not' ('not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ('not' ('not' a.x
    'or' b.x) 'or' (a 'imp' c).x) by BVFUNC_1:def 8
      .=('not' 'not' a.x '&' ('not' 'not' b.x '&' 'not' c.x)) 'or' (('not'
    'not' a.x '&' 'not' b.x) 'or' ('not' a.x 'or' c.x)) by BVFUNC_1:def 8
      .=(a.x '&' (b.x '&' 'not' c.x)) 'or' (((c.x 'or' 'not' a.x) 'or' a.x)
    '&' ((c.x 'or' 'not' a.x) 'or' 'not' b.x)) by XBOOLEAN:9
      .=(a.x '&' (b.x '&' 'not' c.x)) 'or' ((c.x 'or' TRUE) '&' ((c.x 'or'
    'not' a.x) 'or' 'not' b.x)) by A1,BINARITH:11
      .=(a.x '&' (b.x '&' 'not' c.x)) 'or' (TRUE '&' ((c.x 'or' 'not' a.x)
    'or' 'not' b.x)) by BINARITH:10
      .=((c.x 'or' 'not' a.x) 'or' 'not' b.x) 'or' (a.x '&' (b.x '&' 'not' c
    .x)) by MARGREL1:14
      .=(((c.x 'or' 'not' a.x) 'or' 'not' b.x) 'or' a.x) '&' (((c.x 'or'
    'not' a.x) 'or' 'not' b.x) 'or' (b.x '&' 'not' c.x)) by XBOOLEAN:9
      .=(((c.x 'or' 'not' a.x) 'or' a.x) 'or' 'not' b.x) '&' (((c.x 'or'
    'not' a.x) 'or' 'not' b.x) 'or' (b.x '&' 'not' c.x)) by BINARITH:11
      .=((c.x 'or' TRUE) 'or' 'not' b.x) '&' (((c.x 'or' 'not' a.x) 'or'
    'not' b.x) 'or' (b.x '&' 'not' c.x)) by A1,BINARITH:11
      .=(TRUE 'or' 'not' b.x) '&' (((c.x 'or' 'not' a.x) 'or' 'not' b.x)
    'or' (b.x '&' 'not' c.x)) by BINARITH:10
      .=TRUE '&' (((c.x 'or' 'not' a.x) 'or' 'not' b.x) 'or' (b.x '&' 'not'
    c.x)) by BINARITH:10
      .=(((c.x 'or' 'not' a.x) 'or' 'not' b.x) 'or' (b.x '&' 'not' c.x)) by
MARGREL1:14
      .=(((c.x 'or' 'not' a.x) 'or' 'not' b.x) 'or' b.x) '&' (((c.x 'or'
    'not' a.x) 'or' 'not' b.x) 'or' 'not' c.x) by XBOOLEAN:9
      .=((c.x 'or' 'not' a.x) 'or' TRUE) '&' (((c.x 'or' 'not' a.x) 'or'
    'not' b.x) 'or' 'not' c.x) by A3,BINARITH:11
      .=TRUE '&' (((c.x 'or' 'not' a.x) 'or' 'not' b.x) 'or' 'not' c.x) by
BINARITH:10
      .=(('not' b.x 'or' (c.x 'or' 'not' a.x)) 'or' 'not' c.x) by MARGREL1:14
      .=((('not' b.x 'or' 'not' a.x) 'or' c.x) 'or' 'not' c.x) by BINARITH:11
      .=(('not' b.x 'or' 'not' a.x) 'or' TRUE) by A2,BINARITH:11
      .=TRUE by BINARITH:10;
    hence thesis by BVFUNC_1:def 11;
end;
