Consider the following premises. exists x_1: T(x_1) <-> Q_2(x_1). forall x_1: P_1(x_1) -> T(x_1). forall x_1: Q(x_1) or S_3(x_1). forall x_1: Q(x_1) -> T(x_1). forall x_1: S_3(x_1) -> P_1(x_1). Can we infer forall x_1: ~Q_2(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: S_3(x_1) -> P_1(x_1), forall x_1: P_1(x_1) -> T(x_1) we can infer forall x_1: S_3(x_1) -> T(x_1) via universal transitivity. From forall x_1: Q(x_1) -> T(x_1), forall x_1: S_3(x_1) -> T(x_1), forall x_1: Q(x_1) or S_3(x_1) we can infer forall x_1: T(x_1) via universal disjunction elimination. From exists x_1: T(x_1) <-> Q_2(x_1) we can infer exists x_1: T(x_1) -> Q_2(x_1) via existential biconditional elimination. Finally, from exists x_1: T(x_1) -> Q_2(x_1), forall x_1: T(x_1) we can infer exists x_1: Q_2(x_1) via existential modus ponens, which contradicts forall x_1: ~Q_2(x_1). Therefore, the answer is no.