Suppose you prove B from A and later also prove A from B. Useless, right?

Well, no. You’ve proven a wonderful fact. You’ve proven that A and B are equivalent. Each follows, by whatever story you gave, from the other.

What you haven’t accomplished is to reach this entailment back to grounding; to show that A is furthermore a tautology. But this is ok. Not everything in math is about reaching back to the ground in this way. This is not the only way in which to be useful.

Mathematicians are in the business of understanding things. To understand that some A and B both readily entail each other is very frequently clarifying. Depending on the context, this may be a greater boost to understanding than the task of showing one or the other to be entailed from first principles.

Of course, one has to understand what one is doing. If you are a student assigned to prove A from some framework, and you prove A from B and B from A, however hypothetically useful this may be for advancing your understanding of A and B as two guises of the same underlying concept, however much I celebrate the theoretical value of such things, you likely will receive neither an A nor a B on the assignment.

[TODO: Speak also of Wittgenstein’s views on mathematicians’ paranoia about “the hidden contradiction”; speak of the value of inconsistent systems despite ex falso quodlibet]