Suppose that A and B are uniform algebras on compact Hausdorff spaces X and Y, respectively. Let rho, tau : Lambda -> A and S, T : Lambda -> B be mappings on a nonempty set A. Suppose that rho(Lambda), tau(Lambda) and S(Lambda), T(Lambda) are closed under multiplications and contain exp A and exp B respectively and that S(e(1)) is an element of S(Lambda)(-1), T(e(2)) is an element of T(Lambda)(-1) with vertical bar S(e(1))T(e(2))vertical bar = 1 on Ch(B) for some fixed e(1), e(2) is an element of A(1) with rho(e(1)) = tau(e(2)) = 1. If sigma(pi) (S(f)T(g)) boolean AND sigma(pi) (rho(f)tau(g)) not equal phi for all f, g is an element of Lambda and there exists a first-countable dense subset D-B in Ch(B), or a first-countable dense subset D-A in Ch(A), then there exists an algebra isomorphism (S) over tilde : A -> B such that (S) over tilde (p(f)) = S(e(1))S-1(f) and (S) over tilde(tau(f)) = T(e(2))T-1(f) for every f is an element of A.