How can I show that $(R \lor P ) \implies (R \lor Q)$ is equivalent to $R \lor (P \implies Q)$

Question

Using logical equivalencies, how can I show that $(R \lor P ) \implies (R \lor Q)$ is equivalent to $R \lor (P \implies Q)$?

For the left side I get:

$\begin{align}(R \lor P ) \implies (R \lor Q) & \iff \lnot(R \lor P) \lor (R \lor Q)\\ & \iff (\lnot R \land \lnot P) \lor (R \lor Q) \end{align}$

And then I get stuck.

For the right side I just get:

$R \lor (P \implies Q) \iff R \lor (\lnot P \lor Q)$

And then I get stuck.

How can I show that:

$(\lnot R \land \lnot P) \lor (R \lor Q) \iff R \lor (\lnot P \lor Q)$

Answer 1

Bridging your work:

\begin{align} &(\lnot R \land \lnot P) \lor (R \lor Q)\\ &(\lnot R \land \lnot P) \lor R \lor Q\\ &((\lnot R \lor R) \land (\lnot P \lor R)) \lor Q\\ & (\lnot P \lor R) \lor Q\\ & R \lor (\lnot P \lor Q) \end{align}