Consider the following premises. forall x_8: P_5(x_8) -> W(x_8). forall x_8: P_5(x_8). forall x_8: W(x_8) -> S_3(x_8). forall x_8: W(x_8) or ~Q_8(x_8). Can we infer exists x_8: ~S_3(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: W(x_8) or ~Q_8(x_8) we can infer forall x_8: Q_8(x_8) -> W(x_8) via universal rewriting. From forall x_8: P_5(x_8) we can infer forall x_8: P_5(x_8) or Q_8(x_8) via universal addition. From forall x_8: P_5(x_8) -> W(x_8), forall x_8: Q_8(x_8) -> W(x_8), forall x_8: P_5(x_8) or Q_8(x_8) we can infer forall x_8: W(x_8) via universal disjunction elimination. Finally, from forall x_8: W(x_8) -> S_3(x_8), forall x_8: W(x_8) we can infer forall x_8: S_3(x_8) via universal modus ponens, which contradicts exists x_8: ~S_3(x_8). Therefore, the answer is no.