Consider the following premises. exists x_18: P_11(x_18) or P_12(x_18). forall x_18: P_12(x_18) -> P_18(x_18). forall x_18: P_11(x_18) -> Q_18(x_18). forall x_18: P_18(x_18) <-> Q_18(x_18). forall x_18: P_2(x_18). Can we infer exists x_18: P_2(x_18) from them? If possible, name the inference rules being used at each step.
 From forall x_18: P_18(x_18) <-> Q_18(x_18) we can infer forall x_18: P_18(x_18) -> Q_18(x_18) via universal biconditional elimination. From forall x_18: P_12(x_18) -> P_18(x_18), forall x_18: P_18(x_18) -> Q_18(x_18) we can infer forall x_18: P_12(x_18) -> Q_18(x_18) via universal transitivity. From forall x_18: P_11(x_18) -> Q_18(x_18), forall x_18: P_12(x_18) -> Q_18(x_18), exists x_18: P_11(x_18) or P_12(x_18) we can infer exists x_18: Q_18(x_18) via existential disjunction elimination. From forall x_18: P_2(x_18), exists x_18: Q_18(x_18) we can infer exists x_18: P_2(x_18) and Q_18(x_18) via existential conjunction. Finally, from exists x_18: P_2(x_18) and Q_18(x_18) we can infer exists x_18: P_2(x_18) via existential simplification. Therefore, the answer is yes.