This paper presents a framework combining variable abstraction with bounded model checking, in order to prove the counterexamples' absence or establish the counterexamples' existence. A mathematical definition of variable minimal unsatisfiability (VMU) is introduced to drive this abstraction refinement process. The set of variables of VMU formula is a minimal one guaranteeing its unsatisfiability. Furthermore, the authors prove that VMU-driven refinement is valid and minimal by mathematical reasoning. Although the determining problem of VMU is as hard as the well-known problem called minimal unsatisfiability (MU), i.e. D^P-complete, the case study has shown that VMU could be more effective than MU in variable abstraction refinement process.