1. Suppose some proposition S has a proof P, but is such that it cannot
be proven by contradiction.
2. Suppose ~S
3. Reiterate ~S
5. Therefore S
7. Thus ~~S
9. But then S is proven by lines 2-8 through contradiction, which contradicts line 1
10. Therefore there does not exist such a proposition S.
Therefore if anything can be proven it can be proven by contradiction.