Let's assume that $B$ is dependent on $\sigma^2$ and $y$.
I tried to show whether both statements are correct, but I actually don't even know where to start.
$\begin{aligned} & P\left(\beta \mid \sigma^2, B, y\right) P(B)=P\left(\beta, B \mid \sigma^2, y\right) \\ & P\left(\beta \mid \sigma^2, \beta, y\right) P\left(B \mid \sigma^2, y\right)=P\left(\beta, B \mid \sigma^2, y\right)\end{aligned}$
Question: Are both expressions correct? If (not), why?