If you haven’t established that A = B and B = C implies A = C then you haven’t proven that = is an equivalence relation on that space including A, B, and C. Unless you’re going to prove it I would go so far as to call it an abuse of notation to continue using = when you mean some other relation where transitivity does not necessarily hold.
What I mean is that at the time something is proven, it might not be clear that those assumptions are equivalent. Like assuming at the start of a proof that the Axiom of Choice holds vs that Zorn’s lemma holds. That those two proofs are “close” requires that you know it’s already proven that they are equivalent. In that sense, I believe whether proofs should be considered close or not is more subjective than objective, as it depends on the previous knowledge of the reader.
My point above is that one mathematician might assume A=B and another might assume A=C, and both prove the same things. That doesn’t mean they know that always B=C even if it’s true, in which case it’s hard to say whether the proofs are close.
