Prove that $A ∩ (B △ C) = (A △ B) ∩ (A △ C)$ I have to prove that for arbitrary sets $A, B, C$ the following statement is true:

$$A ∩ (B △ C) = (A △ B) ∩ (A △ C).$$

I'm familiar with the property $A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)$ which seems very close to the original problem and yet I can't figure out how to solve it. I'm trying to 

Edit: The operator $△$ is defined as $A △ B = (A \setminus B) ∪ (B \setminus A)$. I'm trying to prove the problem with any known properties of sets and axioms from set theory.

Any suggestions and ideas will be greatly appreciated. 
Thank you!
Your intended identity is false. Let's use this diagram:



So $A=\{1,2,4,5\}$, $B=\{2,3,5,6\}$, and $C=\{4,5,6,7\}$. Substituting into your "identity",

$$A\cap(B\Delta C)=(A\Delta B)\cap(A\Delta C)$$
$$\{1,2,4,5\}\cap(\{2,3,5,6\}\Delta \{4,5,6,7\})
=(\{1,2,4,5\}\Delta \{2,3,5,6\})\cap(\{1,2,4,5\}\Delta \{4,5,6,7\})$$
$$\{1,2,4,5\}\cap\{2,3,4,7\}
=\{1,3,4,6\}\cap \{1,2,6,7\}$$
$$\{2,4\}=\{1,6\}$$

which is clearly false.

I would love to use a shaded Venn diagram, but I'm not sure how to do that in Geogebra. Remember, Venn diagrams are your friends! They can be used to check "identities" and to give ideas on how to prove them.
Hint: $A=B$ means $A\subseteq B$ and $B\subseteq A$.

Also, $A\subseteq B$ means $x\in A\Rightarrow x\in B$.

Let $x\in A\cap (B\triangle C)$.

Then $x\in A\cap (B\setminus C\cup C\setminus B)$

$x\in (A\cap B\setminus C)\cup (A\cap C\setminus B)$

$x\in ((A\cap B\setminus C)\cup A)\cap ((A\cap B\setminus C)\cup C\setminus B)$

and so on.