Set theory: proving of set identities Suppose we have four sets A, B, C and X.
How can one prove the following identity:

$(A \cap B \cap C \cap  \neg X) \cup ( \neg A \cap C) \cup ( \neg B \cap C) \cup (C \cap X)=C$

I tried to apply here any of basic set identities like Distributive Law or Associative Law but it led me to nothing.
Any help would be appreciated, then.
 A: Prahlad Vaidyanathan’s suggestion in the comments is probably the easiest way to go, but an algebraic proof is certainly possible. First note that $$(\neg A\cap C)\cup(\neg B\cap C)=(\neg A\cup\neg B)\cap C$$ by one of the distributive laws. Then $C\cap X=X\cap C$, and
$$\big((\neg A\cup\neg B)\cap C\big)\cup(X\cap C)=\big(\neg A\cup\neg B\cup X\big)\cap C$$
by one of the distributive laws. Thus, the original lefthand side is equal to
$$(A\cap B\cap C\cap\neg X)\cup\big((\neg A\cup\neg B\cup X)\cap C\big)\;.\tag{1}$$
Now use one of the De Morgan laws and the fact that $\neg(\neg Y)=Y$ to see that
$$\neg A\cup\neg B\cup X=\neg(A\cap B\cap\neg X)\;,$$
so that $(1)$ is equal to
$$\Big((A\cap B\cap\neg X)\cap C\Big)\cup\Big(\neg(A\cap B\cap\neg X)\cap C\Big)\;,$$
and use a distributive law to pull out the $C$ and get
$$\Big((A\cap B\cap\neg X)\cup\neg(A\cap B\cap\neg X)\Big)\cap C\;.\tag{2}$$
Finally, use the fact that $Y\cup\neg Y$ is the universal set, which I’ll call $U$, so that $(2)$ is $$U\cap C=C\;.$$
