Prove $E(E(Z|X, Y )|Y ) = E(Z|Y )$.

Question

Let $(X,Y)$ be a random vector.

Prove $E(E(Z|X, Y )|Y ) = E(Z|Y )$.

I'm having a hard time understanding how to get this set up. I'm not sure how to set up the integrals for this. This proof was done in class, but I'm not understanding how to get it started and how to work with the integrals.

I know that there's many similar questions including this: Conditional expectation property proof: $E(E(Z\mid X,Y)\mid X)=E(Z\mid X)$

But I want to prove this from the definition of expectation with the integrals for $E(E(Z|X, Y )|Y )$ as was shown in class. I'm looking for an explanation on how the proof works and how the integrals lead to the result. $E(Z|Y)$