 Gödel-Löb logic

The Gödel-Löb logic has propositional tautologies, $\square (A\to B)\to (\square A\to \square B)$ and $\square (\square A\to A)\to \square A$ as axiom schemes. Its rules are modus ponens rule and the necessitation rule; uniform propositional substitution is built into the schemes. Under an arithmetic realization, each axiom and each rule preserves provability in a theory with the Hilbert-Bernays-Löb derivability conditions and diagonalization. This proves arithmetic soundness by induction on derivations; it is distinct from claiming that every true arithmetic sentence is provable.