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-        ¬(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
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