Consider the following premises. forall x: S_3(x) -> Q_1(x). forall x: Q_1(x) -> S_3(x). exists x: S(x) -> Q_1(x). forall x: ~S_3(x). Can we infer exists x: ~S(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q_1(x) -> S_3(x), forall x: S_3(x) -> Q_1(x) we can infer forall x: Q_1(x) <-> S_3(x) via universal biconditional introduction. From forall x: Q_1(x) <-> S_3(x), forall x: ~S_3(x) we can infer forall x: ~Q_1(x) via universal biconditional elimination. Finally, from exists x: S(x) -> Q_1(x), forall x: ~Q_1(x) we can infer exists x: ~S(x) via existential modus tollens. Therefore, the answer is yes.