Consider the following premises. forall x_5: P_2(x_5) or ~W(x_5). forall x_5: P_5(x_5). forall x_5: P_5(x_5) -> W(x_5). Can we infer exists x_5: ~P_2(x_5) from them?
 From forall x_5: P_5(x_5) -> W(x_5), forall x_5: P_5(x_5) we can infer forall x_5: W(x_5). From forall x_5: P_2(x_5) or ~W(x_5) we can infer forall x_5: W(x_5) -> P_2(x_5). Finally, from forall x_5: W(x_5) -> P_2(x_5), forall x_5: W(x_5) we can infer forall x_5: P_2(x_5), which contradicts exists x_5: ~P_2(x_5). Therefore, the answer is no.