Consider the following premises. forall x: T(x) -> R_4(x). exists x: P_2(x) -> R_4(x). forall x: R_4(x) -> P_2(x). forall x: R_4(x) -> T(x). Can we infer forall x: ~(T(x) -> P_2(x)) from them?
That contradicts the premises. Therefore the answer is no.