# Deriving conditional independence from product rule of probability

I am reading Mathematics for Machine Learning and, in the Summary Statistics and Independence section, the author derives the conditional independence of two random variables given a third RV using the product rule of probability $$p(\boldsymbol{x},\boldsymbol{y}) = p(\boldsymbol{y}\vert\boldsymbol{x})p(\boldsymbol{x})$$ (Eq 6.22) as follow

Now, I don't understand how the author derives Equation 6.56 from the product rule of probability (Eq 6.22).

Can someone please show me the full derivation and explain the intuition (if any) ?

Using conditional probability formulas: \begin{align}p(x,y|z)=\frac{p(x,y,z)}{p(z)}=\frac{p(x|y,z)p(y,z)}{p(z)}=\frac{p(x|y,z)p(y|z)p(z)}{p(z)}=p(x|y,z)p(y|z)\end{align}