Hey, can anyone please help me with this problem:

For all sets A, B, C, prove

If $\displaystyle A-C = \emptyset$ then $\displaystyle (A \cup B) - (B \cap C) = (A-B) \cup (B-C)$

This is what I have done so far, but I seem to be stuck:

$\displaystyle (A \cup B) - (B \cap C)$ Thm $\displaystyle A-B=A \cap B'$

$\displaystyle (A \cup B) \cap (B \cap C)'$ DeMorgans

$\displaystyle (A \cup B) \cap (B' \cup C')$

I can't figure it out from there. I'm guessing it might have something to do with distribution but i'm not sure. Any help is appreciated