Assume we have some conditional pdf given by: $$p(x,y\mid z,c)$$
$$p(x,y\mid z,c)=p(x\mid y,z,c)p(y\mid z,c)$$
What is th reasoning behind this formula? Could you give me step by step explanation? What is exactly joint probability density of $x,y$ given some $z,c$ (what properties do we use to derive above formula)?
Second question (similiar in some way): $$p(x\mid z,c)=\int_\theta p(x,\theta\mid z,c) \, d\theta$$
What properties exactly do we use here? Obviously I have intuition behind these formulas (marginal and conditional probability rules), but I want exact properties which were used in derivation above.