This is a bit late, but here is one example I like:
Theorem. A localization of a regular local ring at a prime ideal is still regular.
One way to prove this is to deduce it from
Theorem. Let $R$ be a local ring. Then the following are equivalent:
$R$ is regular
Every $R$-module has a finite length projective resolution
The residue field has a finite length projective resolution.
(To use it, note that if let $P$ be the prime ideal. Since $R$ is regular, then its residue field $R/P$ has a finite length projective resolution. Now localize--this is exact, so we get a finite length projective $R_P$-projective resolution over of $(R/P)_P$, which is the localization.)residue field of $R_P$)
This stuff is in Chapter 19 of Eisenbud's Commutative Algebra.
It's not clear to me how one would try to prove the first theorem from the definitions of regular.
Edit: fixed some mistakes
