Check if for all sets $A, B, C$ the equality $(A \setminus B) \cap C = (A \cap B) \setminus (B \cap C)$ holds
I tried to solve this, I tested a variety of examples and they all made me think that this equality is valid. Therefore, I attempted to manipulate the definition of these operations but these sets simply don't want to be the equal.
From the Axiom of Extensionality, $$x \in (A \setminus B) \cap C \iff (x\in A \land x \notin B) \land x\in C \iff (x\in A \land x\notin B) \land (x \notin B \land x\in C) \Rightarrow x \in (A\cap C) \setminus (B \cup C) $$ But this is impossible to show this "the other way around" and so this equality seems not to be true. However, I cannot find a counterexample. Any hints?