I don't see why there's a theorem required here. Let our formal system be effectively axiomatized. Then by definition its axioms are a recursive set. So I can just check a proof by starting at the top and verifying that every line is either an axiom or follows from a line above via an inference rule. Both of these two checks are recursive. Hence, proof checking is recursive.
Also, the Wikipedia article doesn't say that Craig's Theorem proves recursively enumerable axiomatizations equivalent to (primitive) recursive axiomatizations. I would find it very surprising if this was a consequence. It only says that a recursively enumerable set of formulas (e.g. the provable sentences in Peano arithmetic, not merely its axioms) can be given a (primitive) recursive axiomatization. That's a very much weaker claim.
Also, the Wikipedia article doesn't say that Craig's Theorem proves recursively enumerable axiomatizations equivalent to (primitive) recursive axiomatizations. I would find it very surprising if this was a consequence. It only says that a recursively enumerable set of formulas (e.g. the provable sentences in Peano arithmetic, not merely its axioms) can be given a (primitive) recursive axiomatization. That's a very much weaker claim.