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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:

1. $R$ is regular

2. Every $R$-module has a finite length projective resolution

3. The residue field has a finite length projective resolution.

(To use it, let $P$ be the prime ideal. Since $R$ is regular, $R/P$ has a finite length projective resolution. Now localize--this is exact, so we get a finite length $R_P$-projective resolution of $(R/P)_P$, which is the 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