Rudin's N-variable stability theorem requires testing, in addition to a multidimensional condition on the distinguished boundary, a single one-variable condition. A generalization of Rudin's theorem is presented in this paper, in which the single-variable condition is replaced with n single-variable conditions where1 leq n leq Nat one's choice. The tradeoff is between simpler and a smaller number of single-variable conditions. The special case n = 1 reduces to the original Rudin theorem. Other special cases reduce to some well-known stability theorems, either with 1 or with N single-variable conditions. An example is provided where choosingn neq 1, N is advantageous from the computational point of view.