If I am trying to prove this:

For all sets A,B,C

(A intersection B)\C = (A\C) intersection (B\C)

If I start with proving X is a subset of Y (L.S. = R.S.), could I do the following?

(xeA ^ xeB) ^ (x/e/C)

Then using distributive law...

(xeA ^ x/e/C) ^ (xeB ^ x/e/C) <=> (A\C) intersection (B\C)

And then for Y is a subset of X...

Use the same distributive rule (backwards)

I'm wondering if this is a justified proof? If not, what would be the correct approach?

Thanks