For $\phi$ a smooth compactly supported function on $\mathbb R^m\times\mathbb R^n$, let $\phi_y=\phi(\cdot,y)$. Let $f$ be a distribution on $\mathbb R^m$. Show that if $\psi(y)=f(\phi_y)$, then $$\int_{\mathbb R^n}\psi(y)\;dy=f(\Phi),$$ where $\Phi(x)=\int_{\mathbb R^n}\phi(x,y)\;dy$.
This looks like some version of the Fubini theorem, as we are exchanging the order of integration over $\mathbb R^n$ with applying the linear functional $f$. In fact, it is basically exactly the statement of Fubini in the case that $f$ is the distribution arising from some locally integrable function.
I am not sure about when $f$ is a general distribution though. I know that the map $\psi$ is smooth, but I'm not sure how this might help.