Consider the following premises. U_3(c_4) -> P_4(c_4). forall y_4: P_4(y_4) <-> U_3(y_4). Can we infer ~(P_4(c_4) -> U_3(c_4)) from them?
 From forall y_4: P_4(y_4) <-> U_3(y_4) we can infer P_4(c_4) <-> U_3(c_4). Finally, from P_4(c_4) <-> U_3(c_4) we can infer P_4(c_4) -> U_3(c_4), which contradicts ~(P_4(c_4) -> U_3(c_4)). Therefore, the answer is no.