The original algorithm for the SAT problem, Variable Elimination Resolution (VER/DP) has exponential space complexity. To tackle that, the backtracking-based DPLL procedure  is used in SAT solvers. We present a combination of two techniques: we use NiVER, a special case of VER, to eliminate some variables in a preprocessing step, and then solve the simplified problem using a DPLL SAT solver. NiVER is a strictly formula size not increasing resolution based preprocessor. In the experiments, NiVER resulted in up to 74% decrease in N (Number of variables), 58% decrease in K (Number of clauses) and 46% decrease in L (Literal count). In many real-life instances, we observed that most of the resolvents for several variables are tautologies. Such variables are removed by NiVER. Hence, despite its simplicity, NiVER does result in easier instances. In case NiVER removable variables are not present, due to very low overhead, the cost of NiVER is insignificant. Empirical results using the state-of-the-art SAT solvers show the usefulness of NiVER. Some instances cannot be solved without NiVER preprocessing. NiVER consistently performs well and hence, can be incorporated into all general purpose SAT solvers.