Why $\forall \varphi\in W_0^{1,2},\int_\Omega \Delta u\varphi=0\implies \Delta u=0$

Question

Let $\Omega \subset \mathbb R^n$ open with Lipschitz boundary. Let $u\in W^{2,2}(\Omega )$. In a course I have that : by the fundamental lemma of variation calculus, if $$\int_\Omega (\Delta u)\varphi=0$$ for all $\varphi\in W_0^{1,2}$ then $\Delta u=0$ a.e.

I would agree if $\varphi\in \mathcal C_c^\infty (\Omega )$, but here $\varphi\in W_0^{1,2}(\Omega )$ not $\mathcal C_c^\infty (\Omega )$. So why is this true ?