We consider the Dirichlet problem for semilinear  elliptic equations on a smooth bounded domain $\Omega$. We assume that $\Omega$ is symmetric about a hyperplane $H$ and convex in the direction orthogonal to $H$. Employing Serrin's result on an overdetermined problem, we show that  any nonzero nonnegative solution is necessarily strictly positive. One can thus apply a well-known result of Gidas, Ni and Nirenberg to conclude that the solution is reflectionally symmetric about $H$ and decreasing  away from the hyperplane in the orthogonal direction.