I am self-studying out of Velleman's "How to Prove It", and am trying to solve exercise 7, part a from section 1.5. The problem is to show that
$$ (P \to Q) \land (Q \to R) = (P \to R) \land ((P \leftrightarrow Q) \lor (R \leftrightarrow Q)) $$
By converting the right-hand side to the equivalent AND/OR statements I was able to simplify the expression to
$$(\lnot P \lor Q) \land (\lnot Q \lor R) \land (\lnot P \lor R)$$
which is equivalent to the left-hand side (confirmed through truth tables, Venn diagrams, and Wolfram Alpha), but I am having trouble showing it analytically. Clearly, the first part of this statement is identical to the left-hand side. But I cannot figure out the analytical steps to show that $\lnot P \lor R$ cancels out of the statement. I've tried distributing the statements into each other completely, but I keeping ending up back at the above expression.
By converting back to conditional statements, the fact that $\lnot P \lor R$ is dispensable becomes intuitive to me. That is, it makes sense that $P \to Q$ and $Q \to R$ already implies that $P \to R$. However, I want to be able to show it analytically based on the Boolean laws and I've hit a wall there. I'm sure I'm missing something simple, but it's driving me crazy. Any hints?