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We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of type dXt = b(t, X)dt + s(t, X)dWt, t = 0, and show that for such equations uniqueness in law is equivalent to joint uniqueness in law for deterministic initial conditions. Here W is a cylindrical Wiener process in a separable Hilbert space U and the equation is considered in a Gelfand triple V. H. E, where H is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of Cherny, who proved these statements for the case of finite-dimensional equations.