Lob's theorem is, of course, if $P(y)$ is a provability predicate for $S$, $S$ diagonalisable, then if $(P(A) \rightarrow A)$ is provable in $S$ then $A$ is provable in $S$. I understand the proof of this.
I am interested in the question:
Suppose $P(A) \rightarrow B$ and $P(B) \rightarrow A$ are provable in $S$. Does it then follow that $A$ and $B$ are both provable in $S$?
Is this true? If so, why?
I've tried proving it via variations on the proof of Lob's Theorem, and by trying to show that $P(A) \rightarrow A$ and $P(B) \rightarrow B$ are provable in $S$ given the assumptions, and thus that I can apply Lob's Theorem to get the conclusion. But I cannot seem to get anything to work.