intersection and union proof

Prove by contradiction or otherwise that (A intersect B)=(A intersect C) and (A union B)=(A union C) if and only if B=C.

SO... I know we have to show this both ways since it's an iff.

Going the first way- Let (A intersect B)=(A intersect C) and (A union B)=(A union C). then for any x in A intersect B, x is in A intersect C. and for any x in A union b, x is in A union C. For x in A intersect B, x is in A and x is in B. for X in A intersect C, x is in A and x is in C. Since x is in A, B, C, and A intersect B equals A intersect C, then sets B and C must be equal.

For A union B and A union C... is where I get a little confused... and also going the other way, letting B=C and proving these two things are true.