# Problems with logical conjunction

There is a problem. I have line like this:
$$x in A backslash(Bbackslash C) leftrightarrow xin(Abackslash B)cup(Acap C)$$
and I tried to prove it from left, but stopped here:
$$xin A backslash(Bbackslash C)) leftrightarrow xin A wedge xnotin (Bbackslash C) leftrightarrow xin Abackslash(xnotin B wedge xin C) leftrightarrow ???$$
and tried from right, bout stopped here:
$$xin(Abackslash B)cup(Acap C) leftrightarrow (xin A wedge x notin B) vee (x in A wedge x in C) leftrightarrow x in A wedge (xnotin B vee xin C) leftrightarrow x in A wedge(x notin B lor x in C) leftrightarrow ???$$
So, to be honest, I’m kinda lost. I tried it two times, but still get different things from both sides. It is supposed to be "truth" and I need to show how to prove it. But I have no idea where is the mistake.

Also other question.
Is it correct to do something like this:
$$(xin A wedge x in B) wedge x notin Cleftrightarrow x in A wedge x notin C wedge x in B wedge x notin C$$
and if yes (or not) based on what? Because I tried to find something to base my prove on, but I can’t find anything specific, like theorem or definition.
If something isn’t clear, please say what, and I will try to clarify.