Proving $(p\to q)\land(p\to r) \equiv p\to(q\land r)$ using logic laws -- short cut or incorrect?

Question

Working through this problem:

Using logic laws, show that the following are logically equivalent: $$(p\to q)\land(p\to r)\qquad\text{and}\qquad p\to(q\land r).$$

The way I did the problem is short and straight forward. I don't see any problems with it but want to be sure its kosher the way I did it.

My solution followed by the book's solution:

Answer 1

Not only is your solution correct IMO, but it is also a much cleaner one than the book offers. I would improve the way your solution is written though (see here for a typesetting guide): \begin{align} (p\to q)\land(p\to r) &\equiv (\neg p\lor q)\land(\neg p\lor r)\tag{Implication identity}\\[0.5em] &\equiv \neg p\lor(q\land r)\tag{Distributive law}\\[0.5em] &\equiv p\to(q\land r)\tag{Implication identity}. \end{align}