Let $X,Y,Z$ be random variables jointly distributed on the same probability space then we know the conditional probability $P(X=a,Y=b\mid Z)$ is a random variable which is $Z$ measurable. Can we say

$$P(X=a,Y=b\mid Z)=P(X=a\mid Y=b,Z)P(Y=b\mid Z)?$$

Moreover is it true to say

$$\operatorname E[P(x=a,Y=b \mid Z)]=P(x=a,Y=b)?$$