Let $u \in \mathscr{D}'(X), v \in \mathscr{D}'(Y)$, and $\phi\in\mathscr{C}_c^\infty(X\times Y)$. Then $$ u(v(\phi(x,y))) = v(u(\phi(x,y))), $$ where $u$ acts on $\phi$ w.r.t. $x$ and $v$ acts on $\phi$ w.r.t. $y$. I have found a similar theorem in chapter 5 of Hörmander's book, but he shows this result by first showing that there exists a unique distribution that coincides with usual the tensor product, and then shows that this product is equal to both sides of the equality above, thus also giving the equality above due to uniqueness.
My question: Is it possible to show this result more directly? That is, without dealing with the tensor product. I can show that both sides of the equality are well-defined but I do have no idea how to show that the equality holds.