We define and study a new restricted consistency notion for bounded arithmetic theories T2j. It is the strongest ∀Π1b-statement over S21 provable in T2j, similar to Con(Gi) in Krajíček and Pudlák, (Z. Math. Logik Grundl. Math. 36 (1990) 29) or RCon(Ti1) in Krajı́ček and Takeuti (Ann. Math. Artificial Intelligence 6 (1992) 107). The advantage of our notion over the others is that can directly be used to construct models of T2j. We apply this by proving preservation theorems for theories of bounded arithmetic of the following well-known kind: The ∀Π1b-separation of bounded arithmetic theories S2i from T2j (1⩽i⩽j) is equivalent to the existence of a model of S2i which does not have a Δ0b-elementary extension to a model of T2j.