Consider the following premises. forall y: P_2(y). forall y: Q(y) -> P_4(y). forall y: U(y) -> P_4(y). exists y: Q(y) or U(y). Can we infer forall y: ~P_2(y) from them? If possible, name the inference rules being used at each step.
That contradicts the premises. Therefore the answer is no.