The issue I have is whether or not it is valid to smush the equations together; whether the equation for p(A|B) is valid in the context of the equation for p(B|A). It may be an issue of intuition mismatch, but it seems analogous to simplifying the equation (1-X)*X^2/(1-X) - the value of 1 is still supposed to be undefined, even after you simplify. Here, we have two "versions" of the same set with disagreeing assigned probabilities.

But your description suggests the issue is that I'm trying to think of the set from the proof p(A|B) as still being there, instead of considering p(A|B) as a specific number; that is, I'm trying to interpret it as a variable whose value remains unresolved. If I consider it in the latter terms, the issue goes away.