Proof by Cases, Equivalences, and Existence Proofs

Write the equivalence $[P \to (Q \vee R)] \equiv [(P \wedge \neg Q) \to R]$, verify it with a truth table or known equivalences, and state what a direct proof of the rewritten form assumes and concludes.

State that the cases must exhaust every possibility for the hypothesis and are proved separately, and write the equivalence $(P \vee Q) \to R \equiv (P \to R) \wedge (Q \to R)$ that licenses the method.

Write the form of an existence theorem, there exists $x$ such that $P(x)$, then record that a constructive proof names or builds the object while a nonconstructive proof shows it must exist without naming it.