Consider the following premises. exists x: P_3(x) <-> S(x). forall x: ~R(x). forall x: S(x) -> Q_3(x). forall x: Q_3(x) -> R(x). Can we infer exists x: ~P_3(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S(x) -> Q_3(x), forall x: Q_3(x) -> R(x) we can infer forall x: S(x) -> R(x) via universal transitivity. From forall x: S(x) -> R(x), forall x: ~R(x) we can infer forall x: ~S(x) via universal modus tollens. Finally, from exists x: P_3(x) <-> S(x), forall x: ~S(x) we can infer exists x: ~P_3(x) via existential biconditional elimination. Therefore, the answer is yes.