Consider the following premises. forall x_31: S(x_31) -> ~Q_21(x_31). forall x_31: Q_31(x_31) or Q_21(x_31). forall x_31: P_31(x_31) <-> Q_31(x_31). exists x_31: ~P_31(x_31). forall x_31: U_4(x_31) <-> W(x_31). forall x_31: W(x_31) -> S(x_31). Can we infer forall x_31: U_4(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: S(x_31) -> ~Q_21(x_31), exists x_31: Q_21(x_31) we can infer exists x_31: ~S(x_31) via existential modus tollens. From forall x_31: W(x_31) -> S(x_31), exists x_31: ~S(x_31) we can infer exists x_31: ~W(x_31) via existential modus tollens. Finally, from forall x_31: U_4(x_31) <-> W(x_31), exists x_31: ~W(x_31) we can infer exists x_31: ~U_4(x_31) via existential biconditional elimination, which contradicts forall x_31: U_4(x_31). Therefore, the answer is no.