reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN holds a 'imp' ('not' a 'eqv'
  'not' a) = I_el(Y)
proof
  let a be Function of Y,BOOLEAN;
  for x being Element of Y holds (a 'imp' ('not' a 'eqv' 'not' a)).x = TRUE
  proof
    let x be Element of Y;
    (a 'imp' ('not' a 'eqv' 'not' a)).x =('not' a 'or' ('not' a 'eqv'
    'not' a)).x by BVFUNC_4:8
      .=('not' a 'or' (('not' a 'imp' 'not' a) '&' ('not' a 'imp' 'not' a)))
    .x by BVFUNC_4:7
      .=('not' a 'or' ('not' 'not' a 'or' 'not' a)).x by BVFUNC_4:8
      .=('not' a 'or' I_el(Y)).x by BVFUNC_4:6
      .=TRUE by BVFUNC_1:10,def 11;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
