Further Conditioning upon already Conditional Expectation

Question

Let's say that $Y$ as a sample space of $\{1,2\}$ and $Z$ has a sample space of $\{3,4\}$.

I know that $E[X]=E[X|Y=1]P(Y=1)+E[X|Y=1]P(Y=2).$

Now suppose I now want to further condition upon $Z$. I know that the final result will have four terms. I'm just not sure how to get there step-by-step notationally in the same way that I arrived at the first conditioning.

Answer 1

E[X]=E[X|Y=1,Z=3]P(Y=1,Z=3)+E[X|Y=2,Z=3]P(Y=2,Z=3)+
$\qquad\qquad\qquad\qquad\qquad\qquad$+E[X|Y=1,Z=4]P(Y=1,Z=4)+E[X|Y=2,Z=4]P(Y=2,Z=4)