Consider the following premises. forall x_1: W(x_1) or P_1(x_1). forall x_1: P_1(x_1) -> Q_1(x_1). exists x_1: W(x_1) -> R_1(x_1). forall x_1: Q_1(x_1) -> R_1(x_1). Can we infer forall x_1: ~R_1(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: P_1(x_1) -> Q_1(x_1), forall x_1: Q_1(x_1) -> R_1(x_1) we can infer forall x_1: P_1(x_1) -> R_1(x_1) via universal transitivity. Finally, from exists x_1: W(x_1) -> R_1(x_1), forall x_1: P_1(x_1) -> R_1(x_1), forall x_1: W(x_1) or P_1(x_1) we can infer exists x_1: R_1(x_1) via existential disjunction elimination, which contradicts forall x_1: ~R_1(x_1). Therefore, the answer is no.