From Wikipedia, the Berry-Essen Inequality is given by $$ \sup_{x\in \mathbb{R}}|\text{Pr}(X\leq z)-\Phi(z)|\leq \frac{C E(|X|^3)}{E(X^2)\sqrt{n}}. $$ From a statistics paper that I am reading, I am seeing a conditional analog version of this $$ \sup_{x\in \mathbb{R}}|\text{Pr}(X\leq z|Y)-\Phi(z)|\leq \frac{C E(|X|^3|Y)}{E(X^2|Y)\sqrt{n}}. $$ In this case, $\text{Pr}(X\leq z|Y)$ is a random distribution and the RHS is now a random variable rather than a number. This is similar to the conditonal Markov Inequality found in this link.
I would be much appreciated if someone could provide theories or references on the derivation of these "conditional" inequalities. As I am unable to see how they continue to hold once you condition on some sub sigma algebras.