Suppose we have real continuous random variables $X,Y,Z$ with $Y$ being independent of $X$ and $Z$. Let $\rho_Y$ be the density of $Y$, similarly for $\rho_{X+Y}$, and $t$ a real number. Then I claim
$$\mathbb{E}[Z | X + Y = t] \rho_{X+Y}(t) = \mathbb{E}[Z \rho_Y(t - X)]$$
My 'proof' is a one-liner:
$$\mathbb{E}[Z | X + Y = t] \rho_{X+Y}(t) = \mathbb{E}[Z \delta_0 (X+Y-t)] = \mathbb{E}[ Z \rho_Y(t-X)]$$
where the last step follows from the fact that $Y$ is independent and thus we can integrate it out first. Of course, this is not rigorous due to the $\delta_0$ term, but how do I formalize this step? Or is it wrong?