Property of symmetric set difference

Let $A \Delta B$ represent the symmetric difference of $A$ and $B$ (i.e. $A \Delta B = (A \cap \bar{B}) \cup (\bar{A} \cap B)$). The following property of $\Delta$ is known: If $A \Delta B = A \Delta C$, then $B = C$. Now, I would like to prove this property only with the aid of the known properties of $\cap$ and $\cup$ (associativity, commutativity, etc.). In other words, I don't want to use the definitions of $\cap$ and $\cup$. I don't want the proof to say something like "Let $x \in A \Delta B \ldots$" etc. Is that possible? I have tried, but I can't find the way. 

$$B=\emptyset\mathbin\triangle B=(A\mathbin\triangle A)\mathbin\triangle B=A\mathbin\triangle(A\mathbin\triangle B)=A\mathbin\triangle(A\mathbin\triangle C)=(A\mathbin\triangle A)\mathbin\triangle C=\emptyset\mathbin\triangle C=C$$