I want to show that the two definitions of convolution $$u * \varphi, \quad u \in \mathcal{D}'(\mathbb{R}^{n}) \,\, \mbox{and}\,\, \varphi \in C^{\infty}_{c}(\mathbb{R}^{n})$$ coincide when I consider $\varphi$ as a function and as a distribution.
Let $\psi \in C^{\infty}_{c}(\mathbb{R}^{n})$, then
\begin{align} (u*\varphi)*\psi (x)&=\int \psi(y)(u*\varphi)(x-y)\, dy\\ &=\int \psi(y) \langle u, \varphi (x-y-\cdot) \rangle \, dy\\ &=\langle u, \int \varphi(x-\cdot-y) \psi(y)\, dy \rangle\\ &=\langle u, (\varphi * \psi)(x-\cdot) \rangle\\ &=u *(\varphi* \psi)(x). \end{align}
Therefore the two definitons coincide.
My problem is
Why can I exchange the integral with $u$ in the third equality?
I would like a good justification. Thanks in advanced.