prove if and only if logic equivalent to ($P\lor Q) \rightarrow (P\land Q $)(using the laws without truth table)

Question

Given that the if and only if defines as being equivalent to $[ p \rightarrow q]\land [q\rightarrow p]$, how to prove it is equivalent to $[ p \lor q]\rightarrow [p\land q]$, I'm new to the logic stuffs, and I feel pretty struggle on it. Need help

Answer 1

You can expand using $[r\implies s] \equiv [s\ \lor \sim r]$.

Answer 2

Assume p or q. Case 1:p true. Then from p->q get q true. so have desired conclusion p and q. Case 2:q true. Similar to case 1.

modus ponens used.

I left out the other direction of the overall iff.