I have three random variables: $x$, $y$, $z$ in $\mathbf{R}$.
I know the following about their distributions: $x \sim \text{unif}[-\infty, \infty]$, $y \sim \mathcal{N}(x, \sigma)$, $z \sim \mathcal{N}(x, \sigma)$, $y-x$ and $z-x$ are independent. 

From this, using Bayes' theorem I infer that 

$x|y \sim \mathcal{N}(y, \sigma)$ and $x|z \sim \mathcal{N}(z, \sigma)$.

What I need to find is conditional density of $y$ given $z$, $f(y|z)$.

Many thanks!