This theorem is a fundamental concept in mathematical logic, stating that any non-constructive proof of a statement in a given language can be transformed into a decision procedure for the language.
However, the catch is that Gödel's Incompleteness Theorem shows that such a procedure cannot exist for any sufficiently large language, rendering the concept of a "simple proof" rather... well, incomplete.
As a result, we're stuck with a bunch of incompleteness, and the only way to deal with it is to invent new languages and proofs that are even more incomplete than the last one!