Let R be a Noetherian normal domain and P a prime ideal. The nth symbolic power P^(n) of P can be geometrically interpreted as the set of regular functions of R that vanish to order n at the generic point of V(P). With this geometric insight, if P\subseteq Q are prime ideals within the non-singular locus of Spec(R), then the Local Zariski-Nagata Theorem states P^(n)\subseteq Q^(n) and translates into a natural criterion for vanishing order along non-singular algebraic sets: functions vanishing to order n at the generic point of V(P) must vanish to order at least n at the generic point of V(Q).

However, when Q is a singular prime, the behavior of vanishing orders becomes less intuitive. Huneke, Katz, and Validashti's Uniform Chevalley Lemma rectifies this scenario by providing a constant C, depending on Q, such that if a function vanishes to order Cn at the generic point of V(P), then it must vanish to order at least n at the generic point of V(Q).

In our exploration of uniformity in Noetherian rings, we introduce the Uniform Izumi-Rees Property, which eliminates the dependency of the constant C on Q in the Uniform Chevalley Lemma. Furthermore, we establish that normal domains essentially of finite type over a field enjoy the Uniform Izumi-Rees Property.