This is because we have that and because from we can always conclude that (and that ).
In our case, this means that from we may conclude that .
Similarly I believe that your converting to is wrong. By De Morgan's Laws you get, instead, that .
(Actually, I feel slightly uneasy about your notation. I'm more the "logical chap" comming from the propositional calculus.)
If I'm not mistaken, the given expression is equivalent to in your notation (or in my notation) - but do the rest of the calculation yourself...