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

theorem
  a 'imp' (a 'nand' b) = 'not' (a '&' b)
proof
  thus a 'imp' (a 'nand' b) ='not' (a '&' a '&' b) by Th54
    .='not' (a '&' b);
end;
