I am having a brain freeze. Could you show the steps to get from line 1 to line 2?
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In general, you can't. The second term in the first line, i.e. $p(\theta|\omega)$ corresponds to second and third terms in the second line, i.e. $p(\theta|\omega)p(\omega)$, which means you need to have $p(D|\theta,\omega)=P(D|\theta)$. It means $D,\omega$ are independent given $\theta$, as also noted in the comments. Probably, in your book, there is context indicating this information, e.g. a Bayes net or an experimental setup. A possible scenario is $D$ is your coin toss experiment, $\theta$ is your head probability of your coins. And $\omega$ is a parameter in the prior of $\theta$.