EDIT: Lest this come up repeatedly, let me expand upon a remark I made in a comment below:
I would be interested to know if RH implies the existence of proofs of a known form in a known system within which one does not assume RH, such that the conclusions of all these proofs conjunctively yield RH.
Actually the answer to my question is "yes" though I find my own example unsatisfying(rather like the Goldbach example):
We can check zeros up to a given magnitude rigorously by known (non-trivial) numerical techniques. RH predicts these tests will come out positive, but of course the testsdon't rely on RH. If we know they all come out positive, that's RH. I find this unsatisfying because the little proofs approximate the whole of RH so badly (compared tohow the Chebyshev estimates really do make one feel one has PNT for all practical purposes). Now a family of zero-free regions that union up to $\sigma >1/2$ where each new zero-free region had strong arithmetic consequences, that would seem interesting.