Given two real vector spaces $V$ and $U$ with subspaces $A, C \subset V$, $A\cap C \neq \{0\}$ and $B, D \subset U$, $B\cap D \neq \{0\}$, is it true that

$$(A\otimes B)\cap (C\otimes D) = (A\cap C)\otimes (B\cap D)$$

and if so, is there a proof available in a paper or textbook? It is stated to be true in a comment to a previous question Tensor Products and Intersections but no proof is given.