Even Wikipedia notes that Cox's Theorem makes another approach possible -- that seems like the place to start looking if you want a mathematical proof. So I think Larks came close to the right question (though it may or may not address your concerns).
Cox and Jaynes show that we can start by requiring probability or the logic of uncertainty to have certain features. For example, our calculations should have a type of consistency such that it shouldn't matter to our final answer if we write P(A∩B) or P(B∩A). This, together with the other requirements, ultimately tells us that:
P(A∩B) = P(B)P(A|B) = P(A)P(B|A)
Which immediately gives us a possible justification for both the Kolmogorov definition and Bayes' Theorem.
From what I understand, the Kolmogorov axioms make no mention of conditional probability. That is simply defined. If I really want to show how probability actually works, I'm not going to argue "by definition". Does anyone know a modified form that uses simpler axioms than P(A|B) = P(A∩B)/P(B)?