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Abbreviate the arithmetical provability predicate by $\square$, with the Hilbert-Bernays-Löb derivability conditions understood. Löb's theorem saysChoose a diagonal sentence $D$ with $S⊢D\leftrightarrow (\square D\to A)$. Internalizing its forward implication gives $S⊢\square D\to \square (\square D\to A)$. Internal modus ponens rule gives $\square (\square D\to A)\to (\square \square D\to \square A)$, while positive introspection gives $\square D\to \square \square D$. Combining these yields $S⊢\square D\to \square A$.

Now use the assumed theorem $\square A\to A$. It follows that $S⊢\square D\to A$, and the reverse direction of the diagonal equivalence gives $S⊢D$. External necessitation rule gives $S⊢\square D$, and the previously derived implication gives $S⊢A$. This proves the theorem from the supplied fixed-point property and the derivability conditions.

The Gödel second incompleteness theorem states that a consistent arithmetic theory in this setting cannot prove its own consistency sentence ${\mathrm{Con}}_S=\neg \square \perp$. If it could, it would prove $\square \perp \to \perp$; applying Löb's theorem with $A=\perp$ would make it inconsistent. Therefore $\boxed{S⊬{\mathrm{Con}}_S}$ when $S$ is consistent.

For the standard Gödel sentence and Rosser sentence, the formalized Rosser unprovability theorem and $G\leftrightarrow \mathrm{Con}(T)$ yield $T⊢G\to R$. If $T+\mathrm{Con}(T)$ is consistent, the converse is unprovable under the usual formalized incompleteness hypotheses. A proof of $R\to G$ would give $\square \neg G\to \square \neg R$. Formalized negative unprovability identifies $\neg \square \neg R$ with $\mathrm{Con}(T)$ and $\neg \square \neg G$ with $\mathrm{Con}(T+\mathrm{Con}(T))$. The strengthened theory would then prove its own consistency, contradicting the Gödel second incompleteness theorem. The consistency assumption belongs to this strictness conclusion.