TITLE: Are there any good nonconstructive "existential metatheorems"?
QUESTION [26 upvotes]: Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is actually to find the proof? When I say "substantially easier," the tediousness of formalizing an informal proof should not be considered. In other words, a rigorous but informal proof doesn't count as a nonconstructive demonstration that a formal proof exists, since there is no essential difficulty in producing a formal proof besides workload. If there is some essential barrier, then there's something wrong either with the proof or with the definition of "formal proof." (Although the concept of "informal proof" is by nature vague, it seems reasonable to say that the procedure transforming a typical informal proof to a formal proof is a primitive recursive computation, even though it's probably impossible to make that precise enough to prove.)
I don't know what such a nonconstructive meta-proof could look like, but I also don't see why one couldn't exist. The only near-example I can think of right now is in the propositional calculus: you can prove that a propositional formula is a theorem by checking its truth table, which does not explicitly provide a deduction from the axioms; however, in most interesting cases, that probably isn't much easier (if at all) than exhibiting a proof, just maybe more mechanical. (Of course, estimating the difficulty of proving or disproving a candidate for a propositional theorem is a huge open problem.) Also, I think there is actually a simple way to convert a truth-table proof into an actual deduction from the axioms of propositional calculus, although I don't remember and didn't retrace all of the details. (Anyway, I'm not even so interested in the propositional calculus for this question, since it's not very expressive.)
Another thing that seems vaguely relevant is Godel's completeness theorem, which states that a formula is a theorem of a first-order theory if and only if it is true in every model of that theory. I don't know in what cases it would be easier, though, to show that some formula were true in every model than it would be just to prove the theorem in the theory.

REPLY [5 votes]: This is almost a result of the type the OP wanted:  In his attempt to prove what is now known as the prime number theorem, Chebyshev used an elementary argument involving a cleverly chosen finite sequence of numerical weights to show that the number of primes up to $x$ is between $0.92 x/\log x$ and $1.11 x/\log x$ for large $x$.  It is natural to ask whether Chebyshev's method can be extended to get closer to the prime number theorem.  It was shown by Rosser and Erdos-Kalmar independently that for any $\varepsilon>0$, there existed a sequence of weights for which the elementary method of Chebyshev showed that the number of primes up to $x$ lay between $(1-\varepsilon) x/\log x$ and $(1+\varepsilon) x/\log x$.  Sending $\varepsilon \to 0$, this would seemingly give an elementary proof of the PNT!  But there is a very important catch: the arguments of Rosser and Erdos-Kalmar use the PNT in the proof (and at the time, there was no known elementary proof of the PNT).  So we have a non-elementary proof that elementary proofs of (arbitrary approximations to) the PNT exist!   See Chapter 9 of 
Harold G. Diamond, MR 670132 Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 553--589.
for further discussion.
As it turns out, this particular result is not quite answering the original question, because it can be made constructive: numerically established zero-free regions of zeta can be converted constructively to elementary proofs of the PNT with an epsilon loss.  But it is at least an example of a non-elementary metatheorem about the existence of elementary proofs.