 Iterated provability collapse implies the original theorem

For standard provability satisfying positive introspection, suppose $T⊢{\square }^{j}A\to A$ for some positive integer $j$. Positive introspection gives $T⊢\square A\to {\square }^{j}A$, hence $T⊢\square A\to A$. Löb's theorem yields $T⊢A$. The same argument applies with an already iterated first-order sentence in place of $A$.