how to prove the convolution formular? let $\overset{\backsim} {g}(x)=g(-x)$; suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove $\int_{E_n}(u*\phi)(x)\psi(x)dx=\int_{E_n}u(x)(\overset{\backsim} {\phi}(x)*\psi(x))dx$ ?

We have \begin{align*} \int_{E_n} (u * \phi)(x)\psi(x)\, dx &= \int_{E_n} \int_{E_n} u(y)\phi(x-y)\, dy\,\psi(x)\, dx\\\ &= \int_{E_n} \int_{E_n} u(y)\phi(x-y)\psi(x)\, dx\, dy\\\ &= \int_{E_n} u(y) \int_{E_n} \overset\backsim\phi(y-x)\psi(x)\,dx\, dy\\\ &= \int_{E_n} u(y)(\overset\backsim\phi * \psi)(y)\, dy \end{align*}