\(x\in A\backslash (A\backslash B)\iff\)
\(x \in A\wedge x\notin A\backslash B\iff \)
\(x \in A\wedge \lnot(x\in A \wedge x\notin B)\iff\)
\(x\in A \wedge ( x \notin A \vee x \in B)\iff \)
\((x\in A\wedge x\notin A)\vee(x\in A\wedge x\in B)\iff\)
\(x\in A\wedge x\in B\iff x\in A\cap B\).