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dlthomas comments on The Cartoon Guide to Löb's Theorem - Less Wrong
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The problem is that while Lob's theorem says that whenever we have ((◻C)->C), C is provable, it does not mean that there is a proof of C from ((◻C)->C). This means that you're use of the deduction theorem is incorrect. You can only say (((◻C)->C)->(◻C)). It is clear that (X->Y)->X does not imply (not X)->X, so your little trick doesn't work.
(X->Y)->X and (X->Y)->Y have the same truth table.
Not when X is false and Y is true.
My bad :-\