Indirect proof. Assume that $B \neq C$. Therefore I assume without loss of generality that $\exists x (x \in B \land x \notin C)$. This leaves us with two possible cases:
$x \in A$. But then $x \notin (A \cup B) \backslash (A \cap B)$ and $x \in (A \cup C) \backslash (A \cap C)$ which contradicts the given equality.
$x \notin A$. But then $x \in (A \cup B) \backslash (A \cap B)$ and $x \notin (A \cup C) \backslash (A \cap C)$ which contradicts the given equality as well.
Therefore $B = C$.
Is this proof complete and correct?