Consider the following premises. forall x_10: P_10(x_10) -> R_5(x_10). forall x_10: W_1(x_10) or U(x_10). forall x_10: R_5(x_10) -> R_1(x_10). forall x_10: P_10(x_10) or ~U(x_10). exists x_10: W_1(x_10) -> R_1(x_10). Can we infer forall x_10: ~R_1(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) or ~U(x_10) we can infer forall x_10: U(x_10) -> P_10(x_10) via universal rewriting. From forall x_10: U(x_10) -> P_10(x_10), forall x_10: P_10(x_10) -> R_5(x_10) we can infer forall x_10: U(x_10) -> R_5(x_10) via universal transitivity. From forall x_10: U(x_10) -> R_5(x_10), forall x_10: R_5(x_10) -> R_1(x_10) we can infer forall x_10: U(x_10) -> R_1(x_10) via universal transitivity. Finally, from exists x_10: W_1(x_10) -> R_1(x_10), forall x_10: U(x_10) -> R_1(x_10), forall x_10: W_1(x_10) or U(x_10) we can infer exists x_10: R_1(x_10) via existential disjunction elimination, which contradicts forall x_10: ~R_1(x_10). Therefore, the answer is no.