Consider the following premises. exists x: P_4(x) -> Q(x). forall x: Q(x) -> P_4(x). forall x: ~P_4(x). Can we infer exists x: ~Q(x) from them?
 From forall x: Q(x) -> P_4(x), exists x: P_4(x) -> Q(x) we can infer exists x: Q(x) <-> P_4(x). From exists x: Q(x) <-> P_4(x) we can infer exists x: Q(x) -> P_4(x). Finally, from exists x: Q(x) -> P_4(x), forall x: ~P_4(x) we can infer exists x: ~Q(x). Therefore, the answer is yes.