Consider the following premises. forall x_2: P_2(x_2) or Q_2(x_2). forall x_2: ~P_2(x_2). exists x_2: P(x_2) -> ~Q_2(x_2). Can we infer forall x_2: P(x_2) from them?
 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). Finally, from exists x_2: P(x_2) -> ~Q_2(x_2), forall x_2: Q_2(x_2) we can infer exists x_2: ~P(x_2), which contradicts forall x_2: P(x_2). Therefore, the answer is no.