Consider the following premises. exists x_31: ~P_31(x_31). forall x_31: Q(x_31) -> ~Q_21(x_31). forall x_31: S_4(x_31) -> Q(x_31). forall x_31: P_31(x_31) <-> Q_31(x_31). forall x_31: Q_31(x_31) or Q_21(x_31). forall x_31: R(x_31) <-> S_4(x_31). Can we infer exists x_31: ~R(x_31) from them? If possible, name the inference rules being used at each step.
 From forall x_31: P_31(x_31) <-> Q_31(x_31), exists x_31: ~P_31(x_31) we can infer exists x_31: ~Q_31(x_31) via existential biconditional elimination. From forall x_31: Q_31(x_31) or Q_21(x_31), exists x_31: ~Q_31(x_31) we can infer exists x_31: Q_21(x_31) via existential disjunctive syllogism. From forall x_31: Q(x_31) -> ~Q_21(x_31), exists x_31: Q_21(x_31) we can infer exists x_31: ~Q(x_31) via existential modus tollens. From forall x_31: S_4(x_31) -> Q(x_31), exists x_31: ~Q(x_31) we can infer exists x_31: ~S_4(x_31) via existential modus tollens. Finally, from forall x_31: R(x_31) <-> S_4(x_31), exists x_31: ~S_4(x_31) we can infer exists x_31: ~R(x_31) via existential biconditional elimination. Therefore, the answer is yes.