Consider the following premises. forall x_8: P_4(x_8). exists x_8: ~P_4(x_8) or R_3(x_8). forall x_8: ~P_8(x_8). forall x_8: P_8(x_8) <-> Q_8(x_8). Can we infer exists x_8: R_3(x_8) 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: P_4(x_8) we can infer forall x_8: P_4(x_8) or Q_8(x_8) via universal addition. From forall x_8: P_4(x_8) or Q_8(x_8), exists x_8: ~P_4(x_8) or R_3(x_8) we can infer exists x_8: Q_8(x_8) or R_3(x_8) via existential resolution. Finally, from exists x_8: Q_8(x_8) or R_3(x_8), forall x_8: ~Q_8(x_8) we can infer exists x_8: R_3(x_8) via existential disjunctive syllogism. Therefore, the answer is yes.