Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$ $A$ and $B$ are any sets, prove that $A \smallsetminus (A \smallsetminus B) = A \cap B.$ This formula makes sense when represented on a Venn diagram, but I am having trouble with proving it mathematically.
I have tried letting $x$ be an element of $A$ and continue from there, but it doesn't seem to work out as a valid proof anyways. 
Could anyone please point me in the right direction?
Many thanks. 

Notice that 
\begin{eqnarray*}
x\in A\setminus(A\setminus B) &\Leftrightarrow& (x\in A)\wedge \neg(x\in A\setminus B)\\
&\Leftrightarrow & (x\in A)\wedge  \neg((x\in A)\wedge \neg (x\in B))\\
&\Leftrightarrow & (x\in A)\wedge ((x\notin A)\lor(x\in B))\\
&\Leftrightarrow & ((x\in A)\wedge (x\notin A))\lor ((x\in A)\wedge(x\in B))\\
&\Leftrightarrow & (x\in A)\wedge (x\in B)\\
&\Leftrightarrow & x\in A\cap B.
\end{eqnarray*}
