reserve Y for non empty set,
  a,b,c,d for Function of Y,BOOLEAN;

theorem
  (a 'imp' b) 'imp' (((a 'imp' c) 'imp' b) 'imp' b) = I_el(Y)
proof
  (a 'imp' b) 'imp' (((a 'imp' c) 'imp' b) 'imp' b) ='not' (a 'imp' b)
  'or' (((a 'imp' c) 'imp' b) 'imp' b) by BVFUNC_4:8
    .='not' ('not' a 'or' b) 'or' (((a 'imp' c) 'imp' b) 'imp' b) by BVFUNC_4:8
    .=('not' 'not' a '&' 'not' b) 'or' (((a 'imp' c) 'imp' b) 'imp' b) by
BVFUNC_1:13
    .=(a '&' 'not' b) 'or' ('not' ((a 'imp' c) 'imp' b) 'or' b) by BVFUNC_4:8
    .=(a '&' 'not' b) 'or' ('not' (('not' a 'or' c) 'imp' b) 'or' b) by
BVFUNC_4:8
    .=(a '&' 'not' b) 'or' ('not' ('not' ('not' a 'or' c) 'or' b) 'or' b) by
BVFUNC_4:8
    .=(a '&' 'not' b) 'or' ('not' (('not' 'not' a '&' 'not' c) 'or' b) 'or'
  b) by BVFUNC_1:13
    .=(a '&' 'not' b) 'or' (('not' (a '&' 'not' c) '&' 'not' b) 'or' b) by
BVFUNC_1:13
    .=(a '&' 'not' b) 'or' ((('not' a 'or' 'not' 'not' c) '&' 'not' b) 'or'
  b) by BVFUNC_1:14
    .=(a '&' 'not' b) 'or' ((('not' a 'or' c) 'or' b) '&' ('not' b 'or' b))
  by BVFUNC_1:11
    .=(a '&' 'not' b) 'or' ((('not' a 'or' c) 'or' b) '&' I_el(Y)) by
BVFUNC_4:6
    .=(a '&' 'not' b) 'or' (('not' a 'or' c) 'or' b) by BVFUNC_1:6
    .=((a '&' 'not' b) 'or' b) 'or' ('not' a 'or' c) by BVFUNC_1:8
    .=((a 'or' b) '&' ('not' b 'or' b)) 'or' ('not' a 'or' c) by BVFUNC_1:11
    .=((a 'or' b) '&' I_el(Y)) 'or' ('not' a 'or' c) by BVFUNC_4:6
    .=(a 'or' b) 'or' ('not' a 'or' c) by BVFUNC_1:6
    .=b 'or' (a 'or' ('not' a 'or' c)) by BVFUNC_1:8
    .=b 'or' ((a 'or' 'not' a) 'or' c) by BVFUNC_1:8
    .=b 'or' (I_el(Y) 'or' c) by BVFUNC_4:6
    .=b 'or' I_el(Y) by BVFUNC_1:10;
  hence thesis by BVFUNC_1:10;
end;
