Weak differentiability and diffeomorphisms

Question

Let $U,V\subset\mathbb{R}^n$ be open sets and assume the existence of a $\mathcal{C}^1$-diffeomorphism $\phi:U\rightarrow V$. Let $u\in W^{1,p}(U)$, $1\leq p\leq\infty$, and define $v=u\circ\phi^{-1}$. Provided that all relevant integrability conditions are fulfilled, how can one prove weak differentiability of $v$? Starting from the basic definition always involves terms like $\varphi\circ\phi$ (using the transformation rule for integrals) where $\varphi\in\mathcal{C}_c^\infty(V)$. But this does not need to be a test function on $U$. Does someone have an idea how to proceed? Thanks very much in advance!