I'm reading through these notes I found online, and I'm having a bit of trouble distinguishing Bayes's Theorem and Bayes's Rule. The central idea of Bayesian Confirmation Theory is that the our credences conform to the axioms of probability calculus, so that our credence that hypotheses H is true is represented by a probability function C(H). Bayes's Theorem, that for two outcomes A,B, P(A|B)=P(B|A)P(A)/P(B), can be derived from the definition of conditional probability alone. Bayes's Rule says that, in light of new evidence E, our updated credence should be C*(H)=C(H|E). The author of the notes linked above claims that Bayes's Rule does not follow from the axioms of the probability calculus, so we need to justify it in some other way.
Imagine for a second that Bayes's Rule did not exist/had never been postulated. Surely the symbol P(A|B) would still have meaning beyond its mathematical definition P(A|B):=P(A AND B)/P(B); all definitions have some motivation behind them. It is the probability of A given we know B has obtained. Back to the world of credences then, if we accept that credences are modelled by probabilities, i.e. that my credence for a hypothesis can be respresented by a function C(H) which behaves like a probability function, then surely along with that comes the idea of conditional credences i.e. C(H|E) where E is some evidence. Isn't Bayes's rule just stating the obvious?