I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex plane, when $A$ has an accumulation point in $D$.
Now let us assume I have a function of two complex variables $f(z, z')$ that is analytic individually in domains $z \in A$ and $z' \in A'$ respectively, when seen as a function of one variable, holding the other fixed. Now let us say that I want to extend this to a larger domain $D \times D'$ and finally evaluate $f(a, b)$ where $a \in D$ and $b \in D'$.
My question is as follows: Does the order in which I go about this process determine my final result? Say I first hold $z$ fixed and find a continuation of the function to domain $D'$, evaluating $f(z, b)$. Now if the resulting expression is one that can be continued to domain $D$ and I plug in $z=a$, will it give me the same result as compared to if I reversed the order of these operations? Essentially in some sense what I am looking for is a sort of uniqueness theorem for analytic continuation for holomorphic functions on $\mathbb{C}^2$.