Consider the following premises. P_1(a_1). forall y_4: U(y_4). forall y_4: U(y_4) -> S(y_4). Can we infer forall y_4: ~(S(y_4) and P_1(y_4)) from them? If possible, name the inference rules being used at each step.
 From forall y_4: U(y_4) -> S(y_4), forall y_4: U(y_4) we can infer forall y_4: S(y_4) via universal modus ponens. From P_1(a_1) we can infer exists y_4: P_1(y_4) via existential generalization. Finally, from forall y_4: S(y_4), exists y_4: P_1(y_4) we can infer exists y_4: S(y_4) and P_1(y_4) via existential conjunction, which contradicts forall y_4: ~(S(y_4) and P_1(y_4)). Therefore, the answer is no.