Thanks for asking! I should add another sentence in that paragraph; the key step, as Kutta and Quill noted, is that "not A" is logically equivalent to "if A, then False", and in particular the statement "2+2=5 is not provable" is logically equivalent to "if 2+2=5 were provable, then 2+2=5", and then we can then run Löb's Theorem.