Consider the following premises. exists x_3: U(x_3) -> W_1(x_3). forall x_3: U(x_3). forall x_3: Q_3(x_3) -> W_1(x_3). forall x_3: P_4(x_3) -> Q_3(x_3). Can we infer exists x_3: W_1(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_4(x_3) -> Q_3(x_3), forall x_3: Q_3(x_3) -> W_1(x_3) we can infer forall x_3: P_4(x_3) -> W_1(x_3) via universal transitivity. From forall x_3: U(x_3) we can infer forall x_3: U(x_3) or P_4(x_3) via universal addition. Finally, from exists x_3: U(x_3) -> W_1(x_3), forall x_3: P_4(x_3) -> W_1(x_3), forall x_3: U(x_3) or P_4(x_3) we can infer exists x_3: W_1(x_3) via existential disjunction elimination. Therefore, the answer is yes.