Consider the following premises. exists y: U(y) -> T(y). forall y: T(y) -> P(y). forall y: P(y) -> R_4(y). forall y: R_4(y) -> U(y). Can we infer forall y: ~(P(y) <-> U(y)) from them?
 From forall y: P(y) -> R_4(y), forall y: R_4(y) -> U(y) we can infer forall y: P(y) -> U(y). From exists y: U(y) -> T(y), forall y: T(y) -> P(y) we can infer exists y: U(y) -> P(y). Finally, from forall y: P(y) -> U(y), exists y: U(y) -> P(y) we can infer exists y: P(y) <-> U(y), which contradicts forall y: ~(P(y) <-> U(y)). Therefore, the answer is no.