Consider the following premises. forall x_2: S_3(x_2) -> P(x_2). forall x_2: T_3(x_2) or U_1(x_2). forall x_2: T_3(x_2) -> P(x_2). forall x_2: U_1(x_2) -> S_3(x_2). Can we infer forall x_2: P(x_2) or W_3(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: U_1(x_2) -> S_3(x_2), forall x_2: S_3(x_2) -> P(x_2) we can infer forall x_2: U_1(x_2) -> P(x_2) via universal transitivity. From forall x_2: T_3(x_2) -> P(x_2), forall x_2: U_1(x_2) -> P(x_2), forall x_2: T_3(x_2) or U_1(x_2) we can infer forall x_2: P(x_2) via universal disjunction elimination. Finally, from forall x_2: P(x_2) we can infer forall x_2: P(x_2) or W_3(x_2) via universal addition. Therefore, the answer is yes.