Theorem on vacuum stability

We consider an extension of the Standard Model with one or more scalar multiplets beyond the Higgs doublet $\mathrm{\Phi }$. The additional scalar multiplets are supposed to carry arbitrary hypercharges. We prove that, in such a model, if the field configuration where only $\mathrm{\Phi }$ has a nonzero vacuum expectation value (VEV) is a local minimum of the potential, then it has a lower value of the potential than any field configuration where both $\mathrm{\Phi }$ and other scalar multiplets have nonzero VEVs.