# Intersection with Set Difference is Set Difference with Intersection/Proof 2

$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$
 $\displaystyle$  $\displaystyle x \in \left({R \setminus S}\right) \cap T$ $\displaystyle$ $\iff$ $\displaystyle \left({x \in R \land x \notin S}\right) \land x \in T$ Definitions of Set Intersection and Set Difference $\displaystyle$ $\iff$ $\displaystyle \left({x \in R \land x \in T}\right) \land x \notin S$ Rules of Commutation and Association $\displaystyle$ $\iff$ $\displaystyle x \in \left({R \cap T}\right) \setminus S$ Definitions of Set Intersection and Set Difference
$\blacksquare$