Here, a proof is requested for the following tautology: $$(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$$
Its pretty easy to prove; nonetheless, I don't find the formula at all intuitive, and I'd be impressed if this has been used to prove things in "real" mathematics.
Question. Are there any theorems $\Phi$ belonging to mathematics proper (e.g. number theory, groups, rings, real analysis, linear algebra, graph theory, etc.) such that $\Phi$ is most easily proved using this tautology?
I fundamentally have no idea how to tag this question.
If you think of $p \vee q$ as equivalent to $\neg p \implies q$ and $\neg p \vee r$ as equivalent to $p \implies r$, this is used all the time, whenever you have a result that has two cases.