In Bell's Boolean-valued Models and Independence Proofs (third edition), the author describes the "forcing over the universe" approach in the first two chapters and how to get corresponding ctms in chapter 4. However, instead of the usual definition M[G]= sigma_G: sigma in Mᵇ, where B is a complete boolean algebra in a ctm M, Mᵇ is the collection of all B-names in M and sigma_G is the evaluation of sigma w.r.t. the generic filter G, the author defines an equivalence relation sigma sim_G tau as big | sigma= tau big | in G and forms the quotient Mᵇ/G, and then spends several pages showing that this model is well-founded. Does the usual definition M[G] work here (of course it does, but I am wondering if there is a quick way to transfer previous results about Mᵇ to M[G])?