Consider the following premises. forall x_4: W_2(x_4) <-> W_3(x_4). forall x_4: ~W_2(x_4). exists x_4: W_3(x_4) -> P(x_4). forall x_4: W_3(x_4) or ~P(x_4). Can we infer exists x_4: ~P(x_4) from them?
 From forall x_4: W_3(x_4) or ~P(x_4) we can infer forall x_4: P(x_4) -> W_3(x_4). From forall x_4: W_2(x_4) <-> W_3(x_4), forall x_4: ~W_2(x_4) we can infer forall x_4: ~W_3(x_4). From exists x_4: W_3(x_4) -> P(x_4), forall x_4: P(x_4) -> W_3(x_4) we can infer exists x_4: W_3(x_4) <-> P(x_4). Finally, from exists x_4: W_3(x_4) <-> P(x_4), forall x_4: ~W_3(x_4) we can infer exists x_4: ~P(x_4). Therefore, the answer is yes.