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.