Let $X,Y,Z$ be sets. If $X\Delta Z=Y$ then $Y\Delta X = Z$ I'm trying to prove the following basic theorem:

Let $X,Y,Z$ be sets. If $X\Delta Z=Y$ then $Y\Delta X = Z$.

I'm familiar with the Symmetric difference definition but every way I go I got stuck. What is the easiest way to prove this theorem?
 A: Note that symmetric difference is commutative and associative i.e. $A\Delta B = B\Delta A$ and $(A\Delta B) \Delta C = A\Delta (B \Delta C)$ for all sets $A,B,C$. Also $A \Delta \emptyset = \emptyset \Delta A = A$ for all sets $A$.
Assume $X \Delta Z = Y$. Then from this assumption and by applying the mentioned properties of $\Delta$ we have
$$
Y \Delta X = (X \Delta Z) \Delta X = X \Delta (Z \Delta X) = X \Delta (X \Delta Z) = (X \Delta X) \Delta Z = \emptyset \Delta Z = Z.
$$
A: Let's use the fact that the symmetric difference operation is defined uning X-OR , and that  ( A w B) ( with w meaning : exclusive OR) is equivalent to : 
~ ( A <--> B)
as a truth table shows it easily. 
I'll use X, Y and Z as sentence symbols meaning "x belongs to X", "...to Y", "...to Z" ( respectively). 
Translated in the language of membership relation, the original statement is : 
(X w Z) <--> Y , which is equivalent to : 
(1) ~ [( X w Z) w Y ]
(2) ~ [ (Y w X ) w Z ]  ( By : commutativity and associativity of X-OR ) 
(3)  (Y w X ) <--> Z 
But (3) is equivalent to what we wanted to arrive at. 
