Bayes' Theorem with multiple random variables I'm reviewing some notes regarding probability, and the section regarding Conditional Probability gives the following example:
$P(X,Y|Z)=\frac{P(Z|X,Y)P(X,Y)}{P(Z)}=\frac{P(Y,Z|X)P(X)}{P(Z)}$
The middle expression is clearly just the application of Bayes' Theorem, but I can't see how the third expression is equal to the second.  Can someone please clarify how the two are equal?

We know
$$P(X,Y)=P(X)P(Y|X)$$
and
$$P(Y,Z|X)=P(Y|X)P(Z|X,Y)$$
(to understand this, note that if you ignore the fact that everything is conditioned on $X$ then it is just like the first example). 
Therefore
\begin{align*}
P(Z|X,Y)P(X,Y)&=P(Z|X,Y)P(X)P(Y|X)\\
&=P(Y,Z|X)P(X)
\end{align*}
Which derives the third expression from the second.
(However I don't have any good intuition for what the third expression means. Does anyone else?)
