Can someone please help with this natural deduction proof?

Question

$(p\lor q)\wedge (p\lor r) \vdash p\lor(q\land r)$

Thanks!

Answer 1

1) Unpack the premise with $\land$E.

2) With $p \lor q$ use $\lor$E: from $p$, we get $p \lor (q \land r)$ by $\lor$I.

3) From $q$, use again $\lor$E with $p \lor r$. Again from $p$ we have $p \lor (q \land r)$.

4) From $r$ we get $q \land r$ by $\land$I and then $p \lor (q \land r)$ by $\lor$I.

In conclusion, we have derived $p \lor (q \land r)$ in all three branches of our derivation and it's done.