  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 the form \begin{align*}
\text{d}X_t=b(t,X)\text{d}t+\sigma(t,X)\text{d}W_t, \,\,\,t\geq 0, \end{align*}
and show that for such equations uniqueness in law is equivalent to joint
uniqueness in law. Here $W$ is a cylindrical Wiener process in a separable
Hilbert space $U$ and the equation is considered in a Gelfand triple $V
\subseteq H \subseteq E$, where $H$ is some separable (infinite-dimensional)
Hilbert space. This generalizes the corresponding results of A. Cherny for the
case of finite-dimensional equations.
