Consider the following premises. forall x_2: P_2(x_2) or Q_2(x_2). exists x_2: S_1(x_2). forall x_2: ~P_2(x_2). Can we infer forall x_2: ~(S_1(x_2) and Q_2(x_2)) from them? If possible, name the inference rules being used at each step.
 From forall x_2: P_2(x_2) or Q_2(x_2), forall x_2: ~P_2(x_2) we can infer forall x_2: Q_2(x_2) via universal disjunctive syllogism. Finally, from exists x_2: S_1(x_2), forall x_2: Q_2(x_2) we can infer exists x_2: S_1(x_2) and Q_2(x_2) via existential conjunction, which contradicts forall x_2: ~(S_1(x_2) and Q_2(x_2)). Therefore, the answer is no.