So I had to prove:
$$ (A \cap B) - C = A \cap (B - C) = B \cap (A - C) $$
Here's my attempt:
$p:$ If $ x \in (A \cap B) - C $ then $ x \in (A \cap B) \land x \notin C$ therefore $ x \in A \land x \in B \land x \notin C$.
$q:$ If $x \in A \cap (B - C)$ then $ x \in A \land x \in B \land x \notin C$
$r:$ If $x \in B \cap (A - C)$ then $x \in B \land x \in A \land x\notin C$
$p = q = r$ then $ (A \cap B) - C = A \cap (B - C) = B \cap (A - C) $
Is this sufficient proof to prove the relation or at least correct? Do I need to add anything else?
It is sufficient and it is correct. You could also just prove that$$(A\cap B)\setminus C=A\cap(B\setminus C)$$and then add that, by the same argument, we also have$$(A\cap B)\setminus C=B\cap(A\setminus C).$$