Consider the following premises. forall y: P_4(y) <-> P(y). forall y: P_4(y). exists y: W_2(y) -> P(y). Can we infer forall y: ~P(y) from them?
 From forall y: P_4(y) <-> P(y) we can infer forall y: P_4(y) -> P(y). From forall y: P_4(y) we can infer forall y: P_4(y) or W_2(y). Finally, from forall y: P_4(y) -> P(y), exists y: W_2(y) -> P(y), forall y: P_4(y) or W_2(y) we can infer exists y: P(y), which contradicts forall y: ~P(y). Therefore, the answer is no.