Consider the following premises. exists x: P_3(x) -> P(x). forall x: R_1(x) or P_3(x). forall x: P(x) -> R_1(x). forall x: R_1(x) -> P(x). Can we infer forall x: ~P(x) from them?
 From forall x: R_1(x) -> P(x), forall x: P(x) -> R_1(x) we can infer forall x: R_1(x) <-> P(x). From forall x: R_1(x) <-> P(x) we can infer forall x: R_1(x) -> P(x). Finally, from forall x: R_1(x) -> P(x), exists x: P_3(x) -> P(x), forall x: R_1(x) or P_3(x) we can infer exists x: P(x), which contradicts forall x: ~P(x). Therefore, the answer is no.