Consider the following premises. forall x_4: P_3(x_4) -> R(x_4). forall x_4: ~P_3(x_4). forall x_4: R(x_4) -> Q_4(x_4). exists x_4: Q_4(x_4) -> P_3(x_4). Can we infer exists x_4: ~R(x_4) from them?
 From forall x_4: R(x_4) -> Q_4(x_4), exists x_4: Q_4(x_4) -> P_3(x_4) we can infer exists x_4: R(x_4) -> P_3(x_4). From exists x_4: R(x_4) -> P_3(x_4), forall x_4: P_3(x_4) -> R(x_4) we can infer exists x_4: R(x_4) <-> P_3(x_4). Finally, from exists x_4: R(x_4) <-> P_3(x_4), forall x_4: ~P_3(x_4) we can infer exists x_4: ~R(x_4). Therefore, the answer is yes.