- ¬(p ∨ (q ∧ ¬r)) ∧ q ≡ (¬p ∧ q) ∧ r : ≡ ¬(p ∨ (q ∧ ¬r)) ∧ q | De Morgan ≡ ¬p ∧ ¬ (q ∧ ¬r) ∧ q | Associative ≡ (¬p ∧ q) ∧ ¬ (q ∧ ¬r) | De Morgan ≡ (¬p ∧ q) ∧ (¬q ∨ r) | Associative ≡ ¬p ∧ (q ∧ (¬q ∨ r)) | Distributive ≡ ¬p ∧ ((q ∧ ¬q) ∨(q ∧ r)) | Contradiction ≡ ¬p ∧ (false ∨(q ∧ r)) | Identity ≡ ¬p ∧ (q ∧ r) | Associative ≡ (¬p ∧ q) ∧ r | RULE EQUIVALENCE SHOWN