Theorem on vacuum stability

We consider an extension of the Standard Model with one or more scalar multiplets beyond the Higgs doublet $Φ$. The additional scalar multiplets are supposed to carry arbitrary hypercharges. We prove that, in such a model, if the field configuration where only $Φ$ 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 $Φ$ and other scalar multiplets have nonzero VEVs.