A property of Weak Solution

$u\in W_0^{1,2}(\Omega)$ is the weak solution to function $-\Delta u=F(\nabla u)$, where $F:\mathbb R^n\to \mathbb R$ is a map satisfying $|F(v)|\le |v|^2$.Prove: For each $\phi\in W_0^{1,2}(\Omega)\cap L^\infty (\Omega)$, we have$$\int_\Omega \nabla u\nabla\phi=\int_\Omega F(\nabla u)\phi.$$

I tried to find a sequence of functions $\{\phi_m\}$ in $\mathcal D(\Omega)$ to approximate $\phi$. The left hand side of the equation can be easily approximated by the one of $\phi_m$, but I had trouble in doing the same thing to the right hand side.