Consider the following premises. forall x_1: W(x_1) <-> R(x_1). exists x_1: P_2(x_1) -> R(x_1). forall x_1: R(x_1) -> P_2(x_1). forall x_1: ~W(x_1). Can we infer exists x_1: ~P_2(x_1) from them?
 From forall x_1: R(x_1) -> P_2(x_1), exists x_1: P_2(x_1) -> R(x_1) we can infer exists x_1: R(x_1) <-> P_2(x_1). From forall x_1: W(x_1) <-> R(x_1), forall x_1: ~W(x_1) we can infer forall x_1: ~R(x_1). Finally, from exists x_1: R(x_1) <-> P_2(x_1), forall x_1: ~R(x_1) we can infer exists x_1: ~P_2(x_1). Therefore, the answer is yes.