Consider the following premises. forall x_13: P_13(x_13). exists x_13: Q_13(x_13) -> W(x_13). forall x_13: P_13(x_13) -> Q_13(x_13). forall x_13: Q_1(x_13). Can we infer exists x_13: Q_1(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> Q_13(x_13), exists x_13: Q_13(x_13) -> W(x_13) we can infer exists x_13: P_13(x_13) -> W(x_13) via existential transitivity. From exists x_13: P_13(x_13) -> W(x_13), forall x_13: P_13(x_13) we can infer exists x_13: W(x_13) via existential modus ponens. From exists x_13: W(x_13), forall x_13: Q_1(x_13) we can infer exists x_13: W(x_13) and Q_1(x_13) via existential conjunction. Finally, from exists x_13: W(x_13) and Q_1(x_13) we can infer exists x_13: Q_1(x_13) via existential simplification. Therefore, the answer is yes.