Consider the following premises. exists x_11: P_11(x_11). forall x_11: P_11(x_11) -> Q_11(x_11). forall x_11: S_1(x_11) -> P_4(x_11). forall x_11: P_4(x_11) -> ~Q_11(x_11). Can we infer exists x_11: ~S_1(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) -> Q_11(x_11), exists x_11: P_11(x_11) we can infer exists x_11: Q_11(x_11) via existential modus ponens. From forall x_11: P_4(x_11) -> ~Q_11(x_11), exists x_11: Q_11(x_11) we can infer exists x_11: ~P_4(x_11) via existential modus tollens. Finally, from forall x_11: S_1(x_11) -> P_4(x_11), exists x_11: ~P_4(x_11) we can infer exists x_11: ~S_1(x_11) via existential modus tollens. Therefore, the answer is yes.