First of all, in practice when we say "Conjecture A is equivalent to Conjecture B," what we mean is "We *have a proof* that Conjecture A is true iff Conjecture B is true." We can have such a proof without having a proof of either conjecture, so this is a meaningful situation. Of course, it will (hopefully) *later become* trivial, when we prove or disprove the conjectures (and so reduce it to "they're both true" or "they're both false").

But your question has more to it than that: suppose we want to say that two *theorems we already know to be true* are equivalent. How can we do that? (Note that this is something we in fact do all the time - e.g. when we say "the compactness of the real numbers is equivalent to their satisfying the extreme value theorem.")

The simplest approach to this is by considering extremely weak axiom systems, which aren't strong enough to prove either result but *can* prove the equivalence. That is, we work over some very weak "base theory."

Historically, of course, the most well-known example is the study of equivalences/implications between versions of the axiom of choice over the theory ZF; as a fun fact, there's a famous story that when Tarski tried to publish a certain equivalence over ZF, one editor rejected it on the grounds that the equivalence between two *true* statements isn't interesting and the other rejected it on the grounds that the equivalence between two *false* statements isn't interesting. *(I believe there were also hints of interest in equivalences between true principles in the study of absolute geometry, but I'm not certain - it's been a while since I looked at the history of non-Euclidean geometry.)* However, ZF is "too strong" for most statements of mathematical interest, so we want to go deeper into things.

This is one of the motivations behind **reverse mathematics**: we look at equivalences/implications/nonimplications over a very weak theory, RCA$_0$, which intuitively corresponds to "computable" mathematics with "finitistic" induction only.

For example, here are some statements which are all equivalent to each other in the sense of reverse mathematics:

*(There is actually a serious issue here which doesn't really crop up when proving equivalences over ZF, namely that we have to figure out how to express the statements we care about in the language of our base theory; this is an issue with weak theories like RCA$_0$ whose language is quite limited. I'm ignoring this issue completely here.)*

And we sometimes want to go weaker still. Equivalences over theories much weaker than RCA$_0$ have been studied, albeit not (in my understanding) as extensively.

1more comment