Prove the following is a tautology I was trying to prove this statement is a tautology without using truth tables. Something doesn't add it here as I keep getting stuck. Take a look please!

For statements, P, Q and R prove that that the statement
  $$ [(P \implies Q) \implies R] \lor [\neg P \lor Q]$$

It is possible to do this without truth tables right?
Here is what I have so far! :)

The statement $(P \implies Q)$ can be collapsed into $(\neg P \lor Q)$. So we can replace the phrase $[(P \implies Q) \implies R]$ with $(\neg P \lor Q) \implies R$.
  Again, we can collapse that expression and get $(P \land \neg Q) \lor R)$. From here I am not sure where to go. There isn't even an $R$ in the expression $[\neg P \lor Q]$ ! Help would be greatly appreciated! Thank you :)

 A: You’ve done much of it. You have
$$\Big((P\land\neg Q)\lor R\Big)\lor(\neg P\lor Q)\;,$$
but you’d be better off backing up a step to
$$\Big(\neg(\neg P\lor Q)\lor R\Big)\lor(\neg P\lor Q)\;.$$
Now rewrite this as
$$\Big(\neg(\neg P\lor Q)\lor(\neg P\lor Q)\Big)\lor R$$
and notice that the big parenthesis is of the form $\neg S\lor S$.
You can instead work directly from what you already have, if you want:
$$\begin{align*}
(P\land\neg Q)\lor(\neg P\lor Q)&\equiv\Big(P\lor(\neg P\lor Q)\Big)\land\Big(\neg Q\lor(\neg P\lor Q)\Big)\\
&\equiv\Big((P\lor\neg P)\lor Q\Big)\land\Big(\neg P\lor(\neg Q\lor Q)\Big)\\
&\equiv(\top\lor Q)\land(\neg P\lor\top)\\
&\equiv\top\land\top\\
&\equiv\top\;.
\end{align*}$$
A: Applying the same equivalence you used in the first transformation:
From $$((\lnot P \lor Q)\rightarrow R) \lor (P \lor Q)$$
You get $$(\color{blue}{\bf\lnot (\lnot P \lor Q)} \lor R ) \lor \color{blue}{\bf (\lnot P \lor Q)}$$
Which is necessarily true because the law of the excluded middle: 
$$\lnot(\lnot P \lor Q) \lor (\lnot P \lor Q)$$ is necessarily true. And $T \lor R$ is necessarily true.
A: You're almost there. 
(¬P∨Q)⟹R = ¬(¬P∨Q)∨R
¬(¬P∨Q)vR∨(¬P∨Q)
Let (¬P∨Q) = A. Then you have ¬A v R v A. So you have NOT A or A. Which is always true.
