Showing $T(\int_{\Omega}f(\cdot,y)dy)=\int_{\Omega}T(f(\cdot,y))dy$

Question

Let $\Omega\subset \mathbb{R}^n$ be an open set, $D(\Omega\times\Omega)=C^{\infty}_c(\Omega\times\Omega)$ (i.e. the smooth function with compact support in $\Omega$) and $T\in D'(\Omega)$ be a distribution, i.e. an element of the dual of $D(\Omega)$. Let $f\in D(\Omega\times \Omega)$ we have $f(\cdot,y)$ (for a fixed $y$) and $\int_{\Omega}f(\cdot,y)dy$ that are test functions. I have to prove that $T(\int_{\Omega}f(\cdot,y)dy)=\int_{\Omega}T(f(\cdot,y))dy$ but I don't really know in which direction to go.