We consider the problem of existential quantifier elimination for Boolean formulas in Conjunctive Normal Form (CNF). We present a new method for solving this problem called Derivation of Dependency-Sequents (DDS). A Dependency-sequent (D-sequent) is used to record that a set of quantified variables is redundant under a partial assignment. We introduce a resolution-like operation called join that produces a new D-sequent from two existing D-sequents. We also show that DDS is compositional, i.e. if our input formula is a conjunction of independent formulas, DDS automatically recognizes and exploits this information. We introduce an algorithm based on DDS and present experimental results demonstrating its potential.