Consider the following premises. forall x_4: Q_2(x_4) -> S(x_4). forall x_4: S(x_4) -> Q_2(x_4). forall x_4: S(x_4) <-> T(x_4). forall x_4: ~T(x_4). s -> Q_2(f). Can we infer ~s from them? If possible, name the inference rules being used at each step.
 From forall x_4: S(x_4) <-> T(x_4), forall x_4: ~T(x_4) we can infer forall x_4: ~S(x_4) via universal biconditional elimination. From forall x_4: Q_2(x_4) -> S(x_4), forall x_4: S(x_4) -> Q_2(x_4) we can infer forall x_4: Q_2(x_4) <-> S(x_4) via universal biconditional introduction. From forall x_4: Q_2(x_4) <-> S(x_4), forall x_4: ~S(x_4) we can infer forall x_4: ~Q_2(x_4) via universal biconditional elimination. From forall x_4: ~Q_2(x_4) we can infer ~Q_2(f) via universal instantiation. Finally, from s -> Q_2(f), ~Q_2(f) we can infer ~s via modus tollens. Therefore, the answer is yes.