In this paper we investigate those extensions of the bimodal provability logic ${\vec CSM}_{0}$ (alias ${\vec PRL}_{1}$ or ${\vec F}^{-})$ which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing ${\vec CSM}_{0}$ are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even (...) complete with respect to Kripke semantics. (shrink)