The marginalization rule of probability for a joint distribution $P(X,\theta)$ is
$$ P(X) = \int P(X,\theta) d\theta$$
on the other hand, the product rule says that $P(X,\theta) = P(X|\theta)P(\theta)$. Combining both
$$ P(X) = \int P(X|\theta)P(\theta) d\theta$$
In the case of having training that, you need to add conditions to the distributions, but nothing changes fundamentally. For notational simplicity, let's call $X$ and $Y$ the training data, $X'$ the point at which you want the prediction and $Y'$ the predicted value. You have a parameter $\theta$. You look for $P(Y'|X',X,Y)$. To find this, we apply the marginalization rule for the parameter $\theta$, thus
$$ P(Y'|X',X,Y) = \int P(Y'|X',X,Y,\theta)P(\theta|X',X,Y) d\theta$$
The last step is to realize that some of the conditions you have are unnecessary. For predicting $Y'$ you have a model that needs only $X'$ and $\theta$, but not the training data $X$,$Y$ explicitly (because the training data is used only for finding $\theta$). Thus, once you have $\theta$, $P(Y'|X',X,Y,\theta) \to P(Y'|X',\theta)$. On the other hand, how you obtain $\theta$ depends only on your training data, and obviously not on which value $X'$ you will want to make a future prediction, thus $P(\theta|X',X,Y) \to P(\theta|X,Y)$. The final expression is thus
$$ P(Y'|X',X,Y) = \int P(Y'|X',\theta)P(\theta|X,Y) d\theta$$
which is what you were looking for except for the change in the notation.
