 Löb's theorem

Assume the Hilbert-Bernays-Löb derivability conditions and the diagonal lemma. Choose $D$ with $T⊢D\leftrightarrow (\square D\to A)$. Necessitating the forward implication and using distribution and positive introspection gives $T⊢\square D\to \square A$. If $T⊢\square A\to A$, then $T⊢\square D\to A$, so the reverse fixed-point implication gives $T⊢D$. Necessitation gives $T⊢\square D$, and the preceding implications give $T⊢A$. No consistency assumption is needed for this implication between provability assertions.

Löb's theorem is a result in mathematical logic, particularly in the area concerning formal systems and provability. It deals with self-referential statements in formal systems and is often discussed in the context of Gödel's incompleteness theorems.