A new method is presented for eliminating existential quantifiers from a Boolean CNF (Conjunctive Normal Form) formula. This new method designs circuits by solving a quantifier elimination problem (QEP) by using derivation of dependency sequents (DDS). This new method begins with a first quantified conjunctive normal form formula, determines dependency sequents for the left branch and the right branch of a branching variable, resolves dependency sequents derived in both branches, and designs a circuit using a current formula having less variables than the first quantified conjunctive normal formula. Utility of this method includes verification, in particular, determining reachability of a state of a design.