Not one of them denied that Baye's Rule could be derived from the very definition of conditional probability (which is a straightforward consequence of the mathematics).

That's a reasonable response to Will's first two comments, but [ETA: not] as a response to his third comment, mentioning Cox's theorem, or my comment, mentioning decision theory. I don't blame you for not knowing whether they had a coherent system of beliefs, but I do blame you for this non sequitur.

ETA: maybe that would be reasonable if you just substituted Cox for Bayes, but only if these frequentists explicitly rejected their contemporary Cox, rather than just ignored him.