We know that

$\displaystyle A \triangle B = (A\backslash B) \cup (B \backslash A)$

and we also know that

$\displaystyle A \triangle B = (A \cup B) \backslash (A \cap B) $

show that

$\displaystyle (A\backslash B) \cup (B \backslash A) = (A \cup B) \backslash (A \cap B) $

using only the following laws: DeMorgan's, Complement, Distributive, Absorption... basically the main set theory laws.

I have attempted to do this but I'm running into some trouble trying to figure out how to move from the union of the two to the difference of the two.