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

There are several routes to a proof, I will list two:
1) You can make a list of all cases. Since you have three variables, there are 8 possibilites for them to have the values true/false.
You can make a table with column titles: $p,q,r,p \to q, \lnot q \lor  p, \dots$ and enter the truth values, then compare the columns for the two expression you want to be equivalent.
2) As indicated by the hint, you can transform all occurences of $\to$ to $\lor$.
Then you can use the distributivity to bring both expression to a normal form, for example to conjunctive normal form http://en.wikipedia.org/wiki/Conjunctive_normal_form
