I don't think this is true because a proof does more than state a conclusion. It establishes a true path from some premises to that conclusion. Sometimes that p

I don't think this is true because a proof does more than state a conclusion. It establishes a true path from some premises to that conclusion. Sometimes that path continues. For example if you had a general constructive proof that there were infinitely many prime numbers it should be a simple matter to alter it a bit and prove the twin prime conjecture wouldn't it? In general a constructive proof and a non-constructive proof of some fact (say proof by contradiction) are fundamentally different in terms of where you can go with the proof.