We present a method for forgetting concept symbols in ontologies specified in the description logic ALCOI. The method is an adaptation and improvement of a second-order quantifier elimination method developed for modal logics and used for computing correspondence properties for modal axioms. It follows an approach exploiting a result of Ackermann adapted to description logics. An important feature inherited from the modal approach is that the inference rules are guided by an ordering compatible with the elimination order of the concept symbols. This provides more control over the inference process and reduces non-determinism, resulting in a smaller search space. The method is extended with a new case splitting inference rule, and several simplification rules. Compared to related forgetting and uniform interpolation methods for description logics, the method can handle inverse roles, nominals and ABoxes. Compared to the modal approach on which it is based, it is more efficient in time and improves the success rates. The method has been implemented in Java using the OWL API. Experimental results show that the order in which the concept symbols are eliminated significantly affects the success rate and efficiency.