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a ∧ T = a {∧ identity}
a ∨ F = a {∨ identity}
a ∧ a = a {∧ idempotence}
a ∨ a = a {∨ idempotence}
a ∧ F = F {∧ null} annihilation
a ∨ T = T {∨ null} absorbtion
a ∧ b = b ∧ a {∧ commutative}
a ∨ a = b ∨ a {∨ commutative}
(a ∧ b) ∧ c = a ∧ (b ∧ c) {∧ associative}
(a ∨ b) ∨ c = a ∨ (b ∨ c) {∨ associative}
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) {∧ distributes over ∨}
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) {∨ distributes over ∧}
¬(a ∧ b) = ¬a ∨ ¬b {DeMorgan's law}
¬(a ∨ b) = ¬a ∧ ¬b {DeMorgan's law}
¬T = F {negate T}
¬F = T {negate F}
¬(¬a) = a {double negation}
a ∧ ¬a = F {∧ complement} NCL
a ∨ ¬a = T {∨ complement} LEM
a → a ∨ b {∨ implication} weakening
a ∧ b → a {∧ implication} ∧El
a ∧ b → b {∧ implication} ∧Er
→ ≡ → ≡ ∨
a → b ≡ ¬b → ¬a ≡ ¬a ∨ b {implication}
¬a → b ≡ ¬b → a ≡ a ∨ b
a → ¬b ≡ b → ¬a ≡ ¬a ∨ ¬b
¬a → ¬b ≡ b → a ≡ a ∨ ¬b
a → b ≡ ¬a ∨ b {implication}
a → b ≡ ¬b → ¬a {contrapositive}
a ∧ (a → b) ⇒ b {Modus Ponens} a, a → b ⊢ b
(a → b) ∧ ¬b ⇒ ¬a {Modus Tollens} a → b, ¬b ⊢ ¬a
(a ∨ b) ∧ ¬a ⇒ b {∨ syllogism} (a ∨ b), ¬a ⊢ b
(a → b) ∧ (b → c) ⇒ a → c {→ chain} transitivity
(a → b) ∧ (c → d) ⇒ (a ∧ c) → (b ∧ d) {→ combination}
(a ∧ b) → c ≡ a → (b → c) {Currying}
(a → b) ∧ (a → ¬b) ≡ ¬a {absurdity}
a ↔ b ≡ (a → b) ∧ (b → a) {equivalence}