Proofs

The contrapositive law

(p ⇒ q) ⇔ (¬q ⇒ ¬p)

Proof by Contradiction

(¬p ⇒ 0) ⇔ p

Equivalence to Truth

(p ≡ 1) ≡ p

Deduction

(E₁ AND E₂ AND · · · AND E_k) ⇒ E

Modus ponens

(p AND (p ⇒ q)) ⇒ q

Resolution

((p + q)(¬p + r)) → (q + r)