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

theorem
  a 'eqv' 'not' a = O_el(Y)
proof
  thus a 'eqv' 'not' a =(a 'imp' 'not' a) '&' ('not' a 'imp' a) by BVFUNC_4:7
    .=('not' a 'or' 'not' a) '&' ('not' a 'imp' a) by BVFUNC_4:8
    .='not' a '&' ('not' 'not' a 'or' a) by BVFUNC_4:8
    .=O_el(Y) by BVFUNC_4:5;
end;
