Consider the following premises. forall x_19: S(x_19) <-> P_19(x_19). exists x_19: P_19(x_19) <-> Q_19(x_19). forall x_19: S(x_19) <-> Q(x_19). forall x_19: ~Q_19(x_19). Can we infer forall x_19: Q(x_19) from them? If possible, name the inference rules being used at each step.
 From exists x_19: P_19(x_19) <-> Q_19(x_19), forall x_19: ~Q_19(x_19) we can infer exists x_19: ~P_19(x_19) via existential biconditional elimination. From forall x_19: S(x_19) <-> P_19(x_19) we can infer forall x_19: S(x_19) -> P_19(x_19) via universal biconditional elimination. From forall x_19: S(x_19) -> P_19(x_19), exists x_19: ~P_19(x_19) we can infer exists x_19: ~S(x_19) via existential modus tollens. Finally, from forall x_19: S(x_19) <-> Q(x_19), exists x_19: ~S(x_19) we can infer exists x_19: ~Q(x_19) via existential biconditional elimination, which contradicts forall x_19: Q(x_19). Therefore, the answer is no.