Consider the following premises. exists z: Q_4(z) -> ~Q(z). forall z: S(z). forall z: S(z) <-> Q(z). Can we infer exists z: ~Q_4(z) from them?
 From forall z: S(z) <-> Q(z) we can infer forall z: S(z) -> Q(z). From forall z: S(z) -> Q(z), forall z: S(z) we can infer forall z: Q(z). Finally, from exists z: Q_4(z) -> ~Q(z), forall z: Q(z) we can infer exists z: ~Q_4(z). Therefore, the answer is yes.