Consider the following premises. exists x_17: P_11(x_17) -> Q_11(x_17). forall x_17: Q_17(x_17). forall x_17: Q_11(x_17) -> P_17(x_17). forall x_17: S_4(x_17) -> P_11(x_17). forall x_17: P_17(x_17) -> P_11(x_17). forall x_17: P_17(x_17) -> ~Q_17(x_17). Can we infer exists x_17: ~S_4(x_17) from them? If possible, name the inference rules being used at each step.
 From forall x_17: P_17(x_17) -> ~Q_17(x_17), forall x_17: Q_17(x_17) we can infer forall x_17: ~P_17(x_17) via universal modus tollens. From exists x_17: P_11(x_17) -> Q_11(x_17), forall x_17: Q_11(x_17) -> P_17(x_17) we can infer exists x_17: P_11(x_17) -> P_17(x_17) via existential transitivity. From exists x_17: P_11(x_17) -> P_17(x_17), forall x_17: P_17(x_17) -> P_11(x_17) we can infer exists x_17: P_11(x_17) <-> P_17(x_17) via existential biconditional introduction. From exists x_17: P_11(x_17) <-> P_17(x_17), forall x_17: ~P_17(x_17) we can infer exists x_17: ~P_11(x_17) via existential biconditional elimination. Finally, from forall x_17: S_4(x_17) -> P_11(x_17), exists x_17: ~P_11(x_17) we can infer exists x_17: ~S_4(x_17) via existential modus tollens. Therefore, the answer is yes.