Consider the following premises. forall x_4: P(x_4) -> R(x_4). forall x_4: P(x_4) -> R(x_4). exists x_4: R(x_4) -> P(x_4). Can we infer forall x_4: ~(R(x_4) <-> P(x_4)) from them?
 From exists x_4: R(x_4) -> P(x_4), forall x_4: P(x_4) -> R(x_4) we can infer exists x_4: R(x_4) <-> P(x_4). From exists x_4: R(x_4) <-> P(x_4) we can infer exists x_4: R(x_4) -> P(x_4). Finally, from exists x_4: R(x_4) -> P(x_4), forall x_4: P(x_4) -> R(x_4) we can infer exists x_4: R(x_4) <-> P(x_4), which contradicts forall x_4: ~(R(x_4) <-> P(x_4)). Therefore, the answer is no.