Consider the following premises. forall x_4: ~P_4(x_4). forall x_4: S(x_4) -> Q_1(x_4). forall x_4: P_4(x_4) <-> S(x_4). forall x_4: Q_1(x_4) -> S(x_4). Can we infer forall x_4: ~Q_1(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P_4(x_4) <-> S(x_4), forall x_4: ~P_4(x_4) we can infer forall x_4: ~S(x_4) via universal biconditional elimination. From forall x_4: S(x_4) -> Q_1(x_4), forall x_4: Q_1(x_4) -> S(x_4) we can infer forall x_4: S(x_4) <-> Q_1(x_4) via universal biconditional introduction. Finally, from forall x_4: S(x_4) <-> Q_1(x_4), forall x_4: ~S(x_4) we can infer forall x_4: ~Q_1(x_4) via universal biconditional elimination. Therefore, the answer is yes.