Consider the following premises. forall y: R(y) -> P_2(y). forall y: P_2(y) -> R(y). forall y: P_2(y) -> R(y). Can we infer exists y: ~(P_2(y) -> R(y)) from them?
 From forall y: R(y) -> P_2(y), forall y: P_2(y) -> R(y) we can infer forall y: R(y) <-> P_2(y). From forall y: R(y) <-> P_2(y) we can infer forall y: R(y) -> P_2(y). From forall y: P_2(y) -> R(y), forall y: R(y) -> P_2(y) we can infer forall y: P_2(y) <-> R(y). Finally, from forall y: P_2(y) <-> R(y) we can infer forall y: P_2(y) -> R(y), which contradicts exists y: ~(P_2(y) -> R(y)). Therefore, the answer is no.