In one of his lectures notes on Gaussian Process Regression, Kilian Weinberger notes that
$$ P(Y\mid D,X) = \int_{\mathbf{w}}P(Y,\mathbf{w} \mid D,X) d\mathbf{w} = \int_{\mathbf{w}} P(Y \mid \mathbf{w}, D,X) P(\mathbf{w} \mid D) d\mathbf{w} $$
Using this equation, he motivates Gaussian Process Regression, stating that multiplying two Gaussians will also produce a Gaussian.
My intuition tells me that the equation above has to do with the definition of conditional probability, marginalization, and perhaps the chain rule of probability. However, I am unable to hash out the intermediary steps to go from $P(Y\mid D,X)$ to the final integral.
I'm aware that
$$ p(a) = \int_{b} p(a, b) \, db $$
from which we can easily deduce something like
$$ p(a \vert c) = \int_{b} p(a, b \vert c) \, db $$
I'm also aware of the chain rule of probability, though I'm not entire sure if this is relevant in deriving the desired conclusion.
$$ p(a, b, c) = p(a \vert b, c) \cdot p(b \vert c) \cdot p(c) $$
Thanks in advance for your help!