  We introduce the notion of \textit{fibered lifted partially hyperbolic
diffeomorphisms} and we prove that any partially hyperbolic diifeomorphism
isotopic to a fibered lifted one where the isotopy take place inside partially
hyperbolic systems is \textit{dynamically coherent.} Moreover we prove some
global stability result: every two partially hyperbolic diffeomorphisms in the
same connected component of a fibered lifted partially hyperbolic
diffeomorphisms, are leaf conjugate.
