Validate my reasoning for this logical equivalence I've basically worked out how to do this question but not sure about my reasoning:
Question:
Show 
1) $(p \rightarrow q) \land (q \rightarrow (\lnot p \lor r))$
is logically equivalent to:
2) $p \rightarrow (q \land r)$
and I am given this equivalence as a hint: $u \rightarrow v$ is logically equivalent to $(\lnot u) \lor v)$
My reasoning:
From statement (1): $(\lnot p \lor r)$ is equivalent to $(p \rightarrow r)$ (By the hint given)
Hence statement (1) becomes: $(p \rightarrow q) \land (q \rightarrow (p \rightarrow r))$
We assume $p$ is true, therefore $q$ is true 
So $p$ also implies $r$
Therefore $p$ implies $q$ and $p$ also implies $r$
Hence $p \rightarrow (q \land r)$
I understand the basic ideas but I'm really confused as to how I can write it all down logically and clearly

Here is yet another way to do this, this time using the Dijkstra-Scholten-Gries-etc. calculational proof format.  We will start with the most complex side, then (as in other answers to this question) expand $P \rightarrow Q$ to $\lnot P \lor Q$, and then simplify and see where that leads us.
\begin{align}
& (p \rightarrow q) \land (q \rightarrow \lnot p \lor r) \\
\equiv & \;\;\;\;\;\text{"expand $\rightarrow$, twice"} \\
& (\lnot p \lor q) \land (\lnot q \lor \lnot p \lor r) \\
\equiv & \;\;\;\;\;\text{"simplify: factor out $\lnot p$, using the fact that $\lor$ distributes over $\land$"} \\
& \lnot p \lor (q \land (\lnot q \lor r)) \\
\equiv & \;\;\;\;\;\text{"simplify: use $q$ in other side of $\land$"} \\
& \lnot p \lor (q \land (\lnot \textrm{true} \lor r)) \\
\equiv & \;\;\;\;\;\text{"simplify"} \\
& \lnot p \lor (q \land r) \\
\equiv & \;\;\;\;\;\text{"reintroduce $\rightarrow$ -- inspired by the shape of our goal"} \\
& p \rightarrow q \land r \\
\end{align}
This proves the equivalence.
