Consider the following premises. forall x_11: S(x_11) <-> S_3(x_11). exists x_11: ~Q_2(x_11). forall x_11: S_3(x_11) -> Q_2(x_11). Can we infer exists x_11: ~S(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: S(x_11) <-> S_3(x_11) we can infer forall x_11: S(x_11) -> S_3(x_11) via universal biconditional elimination. From forall x_11: S_3(x_11) -> Q_2(x_11), exists x_11: ~Q_2(x_11) we can infer exists x_11: ~S_3(x_11) via existential modus tollens. Finally, from forall x_11: S(x_11) -> S_3(x_11), exists x_11: ~S_3(x_11) we can infer exists x_11: ~S(x_11) via existential modus tollens. Therefore, the answer is yes.