logic - Proving two propositions are logically equivalent (without truth table) -
i have prove ~p→(q→r)≡ q→(pvr)
this i've done far:
q→(pvr) ≡(q→p)v(q→r) ≡ ~(q→p)→(q→r) ≡ (q^~p)→(q→r) ≡ q→(~qvr) v ~p→(q→r) ≡ ~qv(~qvr) v ~p→(q→r) ≡ (~qvr)v ~p→(q→r) ≡ (q→r) v [~p→(q→r)]
how should solve this?
~p→(q→r) <=> p v (q→r) <=> p v (~q v r) <=> p v ~q v r q→(p v r) <=> ~q v (p v r) <=> ~q v p v r <=> p v ~q v r
here using rule p→q <=> ~p v q
, fact disjunction associative , commutative.
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