Consider the following premises. forall y_4: Q_2(y_4) -> W_1(y_4). exists y_4: W_1(y_4) -> Q_2(y_4). forall y_4: ~Q_2(y_4). forall y_4: S(y_4) <-> W_1(y_4). Can we infer forall y_4: S(y_4) from them? If possible, name the inference rules being used at each step.
 From exists y_4: W_1(y_4) -> Q_2(y_4), forall y_4: Q_2(y_4) -> W_1(y_4) we can infer exists y_4: W_1(y_4) <-> Q_2(y_4) via existential biconditional introduction. From exists y_4: W_1(y_4) <-> Q_2(y_4), forall y_4: ~Q_2(y_4) we can infer exists y_4: ~W_1(y_4) via existential biconditional elimination. Finally, from forall y_4: S(y_4) <-> W_1(y_4), exists y_4: ~W_1(y_4) we can infer exists y_4: ~S(y_4) via existential biconditional elimination, which contradicts forall y_4: S(y_4). Therefore, the answer is no.