# Show that $(A\cap\bar{B}) \cup (\bar{A}\cap \bar{C}) = (A\cap\bar{B}) \cup (\bar{A}\cap \bar{C}) \cup (\bar{B} \cap \bar{C})$ by rewriting

$A,B,C$ are subsets of the universal set $U$.
$\bar{A}$ is the absolute complement of $A$ = $U\setminus A$

Show that $(A\cap\bar{B}) \cup (\bar{A}\cap \bar{C}) = (A\cap\bar{B}) \cup (\bar{A}\cap \bar{C}) \cup (\bar{B} \cap \bar{C})$ by rewriting either the left hand side or the right hand side.

I have been able to prove the similarity by showing that $\bar{B}\cap \bar{C} \subseteq (A\cap\bar{B}) \cup (\bar{A}\cap \bar{C})$, but I want to show it by only rewriting either the RHS or the LHS by using rules like DeMorgan's laws, Absorption laws etc.

The problem I'm having is that in the RHS $\bar{B}\cap \bar{C}$ is redundant; I already have the two sets that I want and if I try to "get rid of" $\bar{B}\cap \bar{C}$ by taking the union of it and some of the other sets, like $(\bar{B}\cap \bar{C}) \cup (\bar{A}\cap \bar{C})$ then I don't have $(\bar{A}\cap \bar{C})$ on the RHS anymore. But I feel like it must be possible to always rewrite expressions of sets to all equivalent expressions by only using similarity laws like DeMorgan's laws. Maybe I'm wrong.