Prove that ~[r ∨ (q ∧ (~r →~p))] ≡ ~r ∧ (p∨ ~q) by using a series of logical equivalences.

Prove that ~[r ∨ (q ∧ (~r →~p))] ≡ ~r ∧ (p∨ ~q) by using a
series of logical equivalences.

Take RHS ~r ∧ (p ∨~q)

≡ ~r ∧ (~q ∨ p) Commutative law applied

≡ (~r ∧¬q) ∨ (~r ∧ p) Distributive law applied

≡ (~r ∧~q) ∨ ((~r ∧ ~r) ∧ p) Idempotent law applied≡ (~r ∧¬q) ∨ (~r ∧ (~r ∧ p)) Associative law applied

≡ ~r ∧ (~q ∨ (~r ∧ p)) Distributive law applied

≡ ~r ∧ (~q ∨ ~ (r ∨ ~p)) De Morgan’s law & double negation applied

≡ ~r ∧ ~ (q ∧ (r ∨~p)) De Morgan’s law applied

≡ ~r ∧ ~ (q ∧ (~~r ∨ ~p)) Double negation law applied

≡ ~r ∧ ~ (q ∧ (~r → ~p)) Conditional rewritten as disjunction applied≡ ¬[r ∨ (q ∧ (¬r → ¬p))] De Morgan’s law applied

Proved that ~ [r ∨ (q ∧ (~r →~p))] ≡ ~r ∧ (p∨ ~q)