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