Consider the following premises. forall x_18: P_18(x_18) or Q_12(x_18). forall x_18: ~Q_18(x_18). forall z_1: Q_4(z_1). forall x_18: P_18(x_18) <-> Q_18(x_18). Can we infer exists z_1: Q_12(z_1) and Q_4(z_1) 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), forall x_18: ~Q_18(x_18) we can infer forall x_18: ~P_18(x_18) via universal biconditional elimination. From forall x_18: P_18(x_18) or Q_12(x_18), forall x_18: ~P_18(x_18) we can infer forall x_18: Q_12(x_18) via universal disjunctive syllogism. From forall x_18: Q_12(x_18) we can infer Q_12(a_10) via universal instantiation. From Q_12(a_10) we can infer exists z_1: Q_12(z_1) via existential generalization. Finally, from exists z_1: Q_12(z_1), forall z_1: Q_4(z_1) we can infer exists z_1: Q_12(z_1) and Q_4(z_1) via existential conjunction. Therefore, the answer is yes.