# Given random variables $X,Y,Z$, under what conditions is $P(Y|X)=P(Y|X,Z)$?

From this link, where the statement is given for events and not random variables, I gather that for random variables $$X,Y,Z$$, $$P(Y|X)=P(Y|X,Z)$$ only if $$P(Y,Z|X)=P(Y|X)P(Z|X)$$? Does this imply that $$Y$$ and $$Z$$ being conditionally independent (conditioned on $$X$$) is sufficient for $$P(Y|X)=P(Y|X,Z)$$ to hold? Further conditions on the relationship between $$X,Y,Z$$ (such as independence between any of the two or mutual independence of all 3) is not required?

No, it's not required. Being $$Y$$ and $$Z$$ conditionally independent on $$X$$ directly implies $$P(Y|X)=P(Y|X,Z)$$
because $$P(Y|X,Z)=\frac{P(Y,Z|X)}{P(Z|X)}=\frac{\overbrace{P(Y|X)P(Z|X)}^{\text{conditional indep.}}}{P(Z|X)}=P(Y|X)$$