A text on distributed detection (Varshney, 1997) makes a claim on conditional probability identities: If $p\left(u = 0 \vert x, H \right)$ is a conditional density for a binary random variable, given data $x$ and hypothesis $H$, and if $p\left(x \vert H \right)$ is a conditional density for the same data $x$, give hypothesis $H$, the claim is:
\begin{equation} \int dx \cdot p\left(u = 0 \vert x, H \right)p\left(x \vert H \right) = p\left(u = 0 \vert H \right) \end{equation}
I apologize for this (vague?) notation that appears in the book. Notation aside, I wonder if this is the Law of Total Probability, and I just cannot see it? I assume (again, notation unclear) that integration is over all $x$ where the integrand is well defined.