Consider the following premises. forall x_8: ~P_8(x_8). forall x_8: P_8(x_8) <-> Q_8(x_8). forall x_8: Q_8(x_8) or Q_3(x_8). Can we infer exists x_1: Q_3(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) <-> Q_8(x_8), forall x_8: ~P_8(x_8) we can infer forall x_8: ~Q_8(x_8) via universal biconditional elimination. From forall x_8: Q_8(x_8) or Q_3(x_8), forall x_8: ~Q_8(x_8) we can infer forall x_8: Q_3(x_8) via universal disjunctive syllogism. From forall x_8: Q_3(x_8) we can infer Q_3(a_1) via universal instantiation. Finally, from Q_3(a_1) we can infer exists x_1: Q_3(x_1) via existential generalization. Therefore, the answer is yes.