I get the general reason as to why this is True. I'm just having trouble trying to formulate this into a logically correct proof. My approach was:
(1) Assume $A \Delta B = A\Delta C $
(2) Let $x$ be an interger such that $x \subseteq A \Delta B$ and $x \subseteq A \Delta C$,
$x \subseteq A \cup B$ and $x \subseteq A \cup C$
(3) Seperate into cases, where either $x \subseteq A$, or $x \not\subset A$
And this is where I get stuck. I'm not sure if separating into cases is the right approach. Im stuck on linking $A \Delta B = A\Delta C $ with the fact that if they have the same elements after taking the symmetric difference, all the elements of B must also be in C.