Consider the following premises. exists y_4: Q_3(y_4) <-> R(y_4). forall y_4: ~Q_3(y_4). forall y_4: R(y_4) -> Q_3(y_4). Can we infer exists y_4: ~R(y_4) from them? If possible, name the inference rules being used at each step.
 From exists y_4: Q_3(y_4) <-> R(y_4) we can infer exists y_4: Q_3(y_4) -> R(y_4) via existential biconditional elimination. From exists y_4: Q_3(y_4) -> R(y_4), forall y_4: R(y_4) -> Q_3(y_4) we can infer exists y_4: Q_3(y_4) <-> R(y_4) via existential biconditional introduction. Finally, from exists y_4: Q_3(y_4) <-> R(y_4), forall y_4: ~Q_3(y_4) we can infer exists y_4: ~R(y_4) via existential biconditional elimination. Therefore, the answer is yes.