Löb's Theorem roughly states that for any formal system $T$ with Peano Arithmetic and all formulas $P$
If $T$ proves (if $T$ proves $P$ then $P$) then $T$ proves $P$
What happens if we change the inner condition slightly, so that it is assumed the proof is of a certain size, let's say $1$? Consider the following claim:
If $T$ proves (if $T$ proves $P$ in one step then $P$) then $T$ proves $P$
Proving something in one step is equivalent to having it as an axiom. We can also take the contrapositive of the whole thing, yielding
Suppose $T$ doesn't prove $P$. Then $T$ doesn't prove (if $P$ is an axiom of $T$ then $P$)
If that still holds, then it seems as though $T$ is unable to prove the fact that its axioms are true. This actually seems reasonable if we remember that "$P$ is an axiom of $T$" is actually saying something about the string $P$ the "inner" $T$ sees, which the "outer" $T$ doesn't know how to relate to its own (supposed) axiom $P$.
Is the modified claim still a theorem, and, if so, is my analysis in the previous paragraph correct?