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Psy-Kosh comments on Belief in Belief - Less Wrong
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"Those who find this confusing may find it helpful to study mathematical logic, which trains one to make very sharp distinctions between the proposition P, a proof of P, and a proof that P is provable"
This is a bit of a side question, but wouldn't a proof that P is provable be a proof of P? In fact, it sounds like a particularly elegant form of proof.
Hrm... if the system is isn't necessarily trustworthy, then that the system proves that it can prove P doesn't mean that it's actually true that it can prove P, I guess.
EDIT: actually, having it as an explicit axiom "If this proves P, then P" runs you into trouble in any system that has something like Lob's theorem.
("if some specific subset of the rest of this system, (ie, other than this axiom) proves P, then P" can potentially be okay, though)