Prove $U^0+W^0 = (U \cap W)^0$ for subspaces $U,W$ of $V$

Question

I can prove that $U^0+W^0 \subset (U \cap W)^0$, however the other side I find very difficult to prove.

Surely I could take some linear functional $\phi \in (U \cap W)^0$, where $\phi(v) = 0 \space\forall v \in U \cap W$. This does not necessarily mean that $\phi(u) = 0 \space \forall u \in U!$

Am I thinking right?

Note: $X^0$ is the annihilator of $X$