Consider the following premises. exists x_7: Q_7(x_7) -> W(x_7). forall x_7: P_7(x_7) -> Q_7(x_7). forall x_7: ~W(x_7). forall x_7: S_4(x_7) -> P_7(x_7). Can we infer exists x_7: ~S_4(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) -> Q_7(x_7), exists x_7: Q_7(x_7) -> W(x_7) we can infer exists x_7: P_7(x_7) -> W(x_7) via existential transitivity. From forall x_7: S_4(x_7) -> P_7(x_7), exists x_7: P_7(x_7) -> W(x_7) we can infer exists x_7: S_4(x_7) -> W(x_7) via existential transitivity. Finally, from exists x_7: S_4(x_7) -> W(x_7), forall x_7: ~W(x_7) we can infer exists x_7: ~S_4(x_7) via existential modus tollens. Therefore, the answer is yes.