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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.

The Gödel-Löb logic $\mathrm{GL}$ uses propositional variables, $\perp$, implication, and a unary modal operator $\square$. Other Boolean connectives are abbreviations; for example $\neg A$ means $A\to \perp$ and $\top$ means $\perp \to \perp$.

Its axioms are all classical propositional tautologies and the two schemesIts rules are the modus ponens rule and the necessitation rule, $\mathrm{GL}⊢A\Rightarrow \mathrm{GL}⊢\square A$. Uniform propositional substitution is allowed, or equivalently built into the axiom schemes. necessitation rule applies to theorems; a temporary assumption cannot simply be boxed. In provability logic, $\square A$ is read as the formal assertion that $A$ is provable.

Simultaneous propositional substitution givesLet $K=\neg (X\wedge Y)$. The first disjunct equals $K\vee \neg (X\vee Y)$. By De Morgan's laws, $\neg (X\vee Y)=\neg X\wedge \neg Y$ implies $K=\neg X\vee \neg Y$, so absorption reduces that first disjunct to $K$. The second disjunct is $\neg Y\wedge K$ and is also absorbed by $K$. ThereforeThe resulting Boolean function is false only when both $X$ and $Y$ are true, providing a four-row truth table check.

Apply the propositional substitution simultaneously to the atomic variables:By distributivity and the complementary literals $Y,\neg Y$,The other disjunct simplifies using logical implication and absorption:Thus $\phi S\equiv X\vee \neg X\equiv \top$, a tautology.

A propositional substitution maps each atomic propositional variable to a propositional formula and extends recursively through every logical connective. Replacements are simultaneous: a variable appearing inside a replacement formula is not replaced again in the same substitution step. Since propositional logic has no variable-binding quantifiers, this operation has no variable-capture problem.