If $A\cap B = B\cap C$, then $(A-B)\cup C = A\cup (C-B)$.
Are the two statements at the end equivalent?
If $A\cap B = B\cap C$, then $(A-B)\cup C = A\cup (C-B)$.
Are the two statements at the end equivalent?
Suppose that $x\in(A-B)\cup C$.
If $x\in C$ and $x\notin B$ then $x\in C-B$ so $x\in A\cup(C-B)$.
If $x\in C$ and $x\in B$ then $x\in B\cap C=A\cap B$ so $x\in A$. Therefore $x\in A\cup C-B$.
If $x\in A-B$ then $x\in A$ so $x\in A\cup C-B$.
This shows the first inclusion. The statement is symmetric in $A$ and $C$ so switching the role of the two shows inclusion in the other direction.