Consider the following premises. forall x_2: S_1(x_2) or Q(x_2). forall x_2: Q(x_2) -> P_2(x_2). forall x_2: P_2(x_2) or ~S_1(x_2). Can we infer forall x_2: P_2(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: P_2(x_2) or ~S_1(x_2) we can infer forall x_2: S_1(x_2) -> P_2(x_2) via universal rewriting. Finally, from forall x_2: S_1(x_2) -> P_2(x_2), forall x_2: Q(x_2) -> P_2(x_2), forall x_2: S_1(x_2) or Q(x_2) we can infer forall x_2: P_2(x_2) via universal disjunction elimination. Therefore, the answer is yes.