According to wikipedia, the posterior predicative distribution is
$p(\tilde x | X) = \int p(x | \theta) p(\theta | X) d\theta$
How does one reach at the given equality?
$p(\tilde x | X) = \int p(\tilde x,\theta |X) d\theta = \int p(\tilde x | \theta, X) p(\theta| X) d\theta$ where I use that $p(\tilde x, \theta, X) = p(X) p(\theta|X) p(\tilde x|\theta, X)$ but this is not consistent with the given formula.
Sorry I realized that I made a terrible mistake in the original description after looking at the comment and the answer, and now I have corrected.