TITLE: Proving the existence of a proof without actually giving a proof
QUESTION [66 upvotes]: In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. the celebrated probabilistic method and many things found in this thread: What are some things we can prove they must exist, but have no idea what they are?
Now what about proofs themselves? Is there some remote (or not so remote) area of mathematics or logic which contains a concrete theorem that is actually capable of being proved by showing that a proof must exist without actually giving such a proof?
Naively, I imagine that this would require formalizing a proof in such a way that it can be regarded as a mathematical object itself and then maybe showing that the set of such is non-empty. The only thing in this direction that I've seen so far is the "category of proofs", but that's not the answer.
This may sound like mathematical science fiction, but initially so does e.g. the idea of proving that some statement is unprovable in a framework, which has become standard in axiomatic set theory.
Feel free to change my speculative tags.

REPLY [3 votes]: Various good answers here, but the following seems to have been left out and is surely the simplest.
Consider the propositional calculus.  (If you are not familiar with this term: essentially it is logic using basic terms like "and", "or", "not", but which does not have quantifiers "for all" or "there exists".)
One can determine whether or not a statement of propositional calculus is "always true" (technically, a tautology) by constructing a truth table.
One can also define proofs in such a system; a theorem is any statement which can be proved.
However, it turns out that tautologies and theorems are in fact the same.  So, by constructing a truth table and proving that a statement is a tautology, you have also proved that it has a proof, but you have not given the proof.
I think this gives an answer to your question, though it should also be remarked that there is an algorithm which in effect "turns a truth table into a proof".  So although in the situation I have described you would not have written down a proof, you would be able to do so by purely mechanical procedures.