Consider the following premises. forall y_1: ~Q(y_1). exists y_1: R_4(y_1) <-> Q(y_1). forall y_1: W(y_1) -> R_4(y_1). forall y_1: R_4(y_1) -> W(y_1). Can we infer exists y_1: ~W(y_1) from them? If possible, name the inference rules being used at each step.
 From exists y_1: R_4(y_1) <-> Q(y_1), forall y_1: ~Q(y_1) we can infer exists y_1: ~R_4(y_1) via existential biconditional elimination. From forall y_1: W(y_1) -> R_4(y_1), forall y_1: R_4(y_1) -> W(y_1) we can infer forall y_1: W(y_1) <-> R_4(y_1) via universal biconditional introduction. Finally, from forall y_1: W(y_1) <-> R_4(y_1), exists y_1: ~R_4(y_1) we can infer exists y_1: ~W(y_1) via existential biconditional elimination. Therefore, the answer is yes.