Understanding Bayes rule

Question

I know Bayes rule as:

$$P(a \cap b) = P(a \mid b)P(b)$$

But I came across:

$$P( a,z \mid b) = P(a \mid z,b) P(z \mid b)$$

How is the last equation proven?

Answer 1

$$P(a, z \mid b) = \dfrac{P(a, z, b)}{P(b)}$$ $$P(a \mid z, b) = \dfrac{P(a, z, b)}{P(z, b)}$$ $$P(z \mid b) = \dfrac{P(z, b)}{P(b)}$$

Answer 2

Assuming $P(C)>0$ and $P(B\cap C)>0$,

$P(A\cap B\cap C)=P(A\mid B\cap C)P(B\cap C)=P(A\mid B\cap C)P(B\mid C)P(C)$

Again, $P(A\cap B\cap C)=P(A\cap B\mid C)P(C)$

Hence $P(A\cap B\mid C)=P(A\mid B\cap C)P(B\mid C)$

Notation might be different, but I believe you can understand that easily!