In Kevin Murphy's Conjugate Bayesian analysis of the Gaussian distribution, he writes that the posterior predictive distribution is
$$ p(x \mid D) = \int p(x \mid \theta) p(\theta \mid D) d \theta $$
where $D$ is the data on which the model is fit and $x$ is unseen data. What I don't understand is why the dependence on $D$ disappears in the first term in the integral. Using basic rules of probability, I would have expected:
$$ \begin{align} p(a) &= \int p(a \mid c) p(c) dc \\ p(a \mid b) &= \int p(a \mid c, b) p(c \mid b) dc \\ &\downarrow \\ p(x \mid D) &= \int \overbrace{p(x \mid \theta, D)}^{\star} p(\theta \mid D) d \theta \end{align} $$
Question: Why does the dependence on $D$ in term $\star$ disappear?
For what it's worth, I've seen this kind of formulation (dropping variables in conditionals) other places. For example, in Ryan Adam's Bayesian Online Changepoint Detection, he writes the posterior predictive as
$$ p(x_{t+1} \mid r_t) = \int p(x_{t+1} \mid \theta) p(\theta \mid r_{t}, x_{t}) d \theta $$
where again, since $D = \{x_t, r_t\}$, I would have expected
$$ p(x_{t+1} \mid x_t, r_t) = \int p(x_{t+1} \mid \theta, x_t, r_t) p(\theta \mid r_{t}, x_{t}) d \theta $$