So far I have that $a|b$ implies $b=ax$ for some x in the integers and $c|d$ implies that $d=cy$ for some y in the integers. From here I can see that $gcd(a,c)|gcd(b,d)$ is logically equivalent to $gcd(a,c)|gcd(ax,cy)$ but I am not sure where to go from there.
If $a\mid b$ and $c\mid d$, then, since $\gcd(a,c)\mid a$ and $a\mid b$, $\gcd(a,c)\mid b$. For the same reason, $\gcd(a,c)\mid d$. So, since $\gcd(a,c)$ divides both $b$ and $d$, it divides $\gcd(b,d)$.