Subspace of Vectors Lemma

Question

Let $A,B,C$ be subspaces of a vector space $V$.

$V = (A\cap(B+C)) \cap (C \cap (B + A)) = A \cap C$. Is this always true ?

I know that I can't use distributivity of intersection with respect to addition as that isn't true for all vector spaces. I am thinking the statement is true.

Answer 1

(1) First assume $x\notin A\cap C$ notice this implies $x \notin A\cap(B+C)$ or $x \notin C\cap(B+A)$, this shows us that if $x\notin A\cap C$, then $x\notin V$.

(2) Then assume $x\in A\cap C$ this implies $x \in A\cap(B+C)$ and $x \in C\cap(B+A)$

(1) and (2) together imply that $V = A\cap C$