One example is Solovay's theorem that if the axiom of determinacy holds, then each subset of $\omega_1$ is constructible from a real. The proof breaks into cases. Case One is when $\omega^{L[t]}_1=\omega_1$ holds for some real $t$. Case Two is when this does not hold. Case One has a direct proof, and then Case Two is reduced to Case One via forcing. The punchline is that Case One never holds!