Consider the following premises. forall x: W(x) <-> P_1(x). exists x: W(x). Can we infer forall x: ~P_1(x) from them?
 From forall x: W(x) <-> P_1(x) we can infer forall x: W(x) -> P_1(x). Finally, from forall x: W(x) -> P_1(x), exists x: W(x) we can infer exists x: P_1(x), which contradicts forall x: ~P_1(x). Therefore, the answer is no.