We give a different and possibly more accessible proof of a general Borsuk-Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb Z/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(\mathbb Z/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk-Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.