Consider the following premises. forall x_1: R(x_1) or ~P_4(x_1). exists x_1: P(x_1) or ~R(x_1). forall x_1: P(x_1) -> P_4(x_1). Can we infer forall x_1: ~(P_4(x_1) <-> R(x_1)) from them?
 From exists x_1: P(x_1) or ~R(x_1) we can infer exists x_1: R(x_1) -> P(x_1). From exists x_1: R(x_1) -> P(x_1), forall x_1: P(x_1) -> P_4(x_1) we can infer exists x_1: R(x_1) -> P_4(x_1). From forall x_1: R(x_1) or ~P_4(x_1) we can infer forall x_1: P_4(x_1) -> R(x_1). Finally, from forall x_1: P_4(x_1) -> R(x_1), exists x_1: R(x_1) -> P_4(x_1) we can infer exists x_1: P_4(x_1) <-> R(x_1), which contradicts forall x_1: ~(P_4(x_1) <-> R(x_1)). Therefore, the answer is no.