Prove$$ A\triangle(B\backslash C)=(A\triangle B)\backslash C $$
If and only if $A\cap C=\emptyset$.

I should prove it using just **or** and **and**. Like this:  

The first inclusion:
$$ x \in A\triangle(B\backslash C)\\
 [x \in A \backslash(B \backslash C)] \vee [x\in(B \backslash C) \backslash A]$$
I think the next step is **wrong**:  

$$
 [x \in (A \backslash B) \backslash C)] \vee [x\in(B \backslash A) \backslash C]\\
 [x \in (A \backslash B) \vee x\in(B \backslash A) ] \land x\notin C\\
 (A\triangle B)\backslash C
 $$