applying Lob theorem For the understanding Lob theorem, I want show an example. Let followings be some famous theorems in mathematics academia for a theory T. They are already proven decades ago. > Theorem 1: if F, then A → B > > Theorem 2: (if T ⊢ ( A → B ) ) then ( C → E ) > > Theorem 3: if ( C → E ), then F Someday, one person found a fact accidentally that ( Theorem 2 and 3 and 1 ) ⊢ { ( if T ⊢ ( A → B ), then ( A → B ) } We already know that Theorem 1, 2, 3 is provable sentence in theory T. So, This person insists that by this fact, T ⊢ ( A → B ) by Lob theorem. Is it correct?

Yes, but if you're trying to use Löb's theorem to resolve a major unsolved problem, then I bet you that one or more of the steps you've just written down is wrong.