Verify these logical equivalences by writing an equivalence proof? I have two parts to this question - I need to verify each of the following by writing an equivalence proof:


*

*$p \to (q \land r) \equiv (p \to q) \land (p \to r)$

*$(p \to q) \land (p \lor q) \equiv q$
Thank you if you can help!  It's greatly appreciated.

We make extensive use of the identity $(a \to b) \equiv (\lnot a \lor b)$, and leave you to fill in the reasons for some of the intermediate steps in (2).
(1) $\quad p \to (q \wedge r) \equiv \lnot p \lor (q \land r) \equiv (\lnot p \lor q) \land (\lnot p \lor r) \equiv (p \to q) \wedge (p \to r)$. 
(2) $\quad(p \to q) \land (p \lor q) \equiv  q\quad?$
$$(p \to q) \land (p \lor q) \equiv (\lnot p \lor q) \land (p \lor q)\tag{1}$$ 
$$\equiv \;[(\lnot p \lor q) \land p] \lor [(\lnot p \lor q) \land q]\tag{2}$$
$$\equiv \;[(\lnot p \land p) \lor (q \land p)] \lor [(\lnot p \land q) \lor (q \land q)]\tag{3}$$
$$\equiv \;F \lor (p \land q) \lor  [(\lnot p \land q) \lor (q \land q)]\tag{4}$$
$$\equiv \;(p\land q) \lor [(\lnot p \land q) \lor (q \land q)]\tag{5}$$
$$\equiv \;(p\land q) \lor (\lnot p \land q) \lor q\tag{6}$$
$$\equiv \;[(p \lor \lnot p) \land q] \lor q\tag{7}$$
$$\equiv \;(T \land q) \lor q\tag{8}$$
$$\equiv \;q\lor q\tag{9}$$
$$\equiv \;q\tag{10}$$
