Consider the following premises. forall x_8: ~P_3(x_8). exists x_8: R_3(x_8) <-> S(x_8). forall x_8: P_3(x_8) <-> S(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 exists x_8: R_3(x_8) <-> S(x_8) we can infer exists x_8: R_3(x_8) -> S(x_8) via existential biconditional elimination. From forall x_8: P_3(x_8) <-> S(x_8), forall x_8: ~P_3(x_8) we can infer forall x_8: ~S(x_8) via universal biconditional elimination. Finally, from exists x_8: R_3(x_8) -> S(x_8), forall x_8: ~S(x_8) we can infer exists x_8: ~R_3(x_8) via existential modus tollens. Therefore, the answer is yes.