Three sets A, B, C are such that $A =B∩C$ and $B=C∩A$, then

$A⊂ B$ $B⊂A$ $A= B$ $A⊂ B'$

$A= B$

We have, $A =B∩C$ and $B=C∩A$ $⇒A=(C∩A)∩C⇒A=C∩A=B$