Laws of logical equivalence

Question

How do I solve this to show that the L.H.S = R.H.S

((p → q) ∨ (¬p → r)) → (q ∨ r) ≡ q ∨ r

I have to show this using the laws of logical equivalence.

I have made some attempt using implication law, associative law and commutative law, but I am not sure if these are the right laws and I am getting a bit confused. Help to solve this would be appreciated.

Answer 1

I have made some attempt using implication law, associative law and commutative law, but I am not sure if these are the right laws and I am getting a bit confused.

Yes, that is correct.  Begin by using implication equivalence on the antecedent's implications, then associate and commute that disjunction.

$$\begin{align}&((p\to q)\vee(\neg p\to r))\to (q\vee r)\\[2ex]&((\neg p\vee q)\vee(p\vee r))\to(q\vee r)\\[2ex]&((\neg p\vee p)\vee(q\vee r))\to(q\vee r)\end{align}$$

Now you should see the next move.