Consider the following premises. forall x_12: P(x_12) or ~R(x_12). exists x_12: P_12(x_12) or ~P(x_12). forall x_12: P_12(x_12) -> R(x_12). Can we infer forall x_12: ~(R(x_12) <-> P(x_12)) from them?
 From exists x_12: P_12(x_12) or ~P(x_12) we can infer exists x_12: P(x_12) -> P_12(x_12). From exists x_12: P(x_12) -> P_12(x_12), forall x_12: P_12(x_12) -> R(x_12) we can infer exists x_12: P(x_12) -> R(x_12). From forall x_12: P(x_12) or ~R(x_12) we can infer forall x_12: R(x_12) -> P(x_12). Finally, from forall x_12: R(x_12) -> P(x_12), exists x_12: P(x_12) -> R(x_12) we can infer exists x_12: R(x_12) <-> P(x_12), which contradicts forall x_12: ~(R(x_12) <-> P(x_12)). Therefore, the answer is no.