Consider the following premises. forall z: ~Q(z). forall z: Q_1(z) -> Q(z). exists z: Q_4(z) -> Q_1(z). forall z: Q(z) -> Q_1(z). Can we infer exists z: ~Q_4(z) from them?
 From forall z: Q(z) -> Q_1(z), forall z: Q_1(z) -> Q(z) we can infer forall z: Q(z) <-> Q_1(z). From forall z: Q(z) <-> Q_1(z), forall z: ~Q(z) we can infer forall z: ~Q_1(z). Finally, from exists z: Q_4(z) -> Q_1(z), forall z: ~Q_1(z) we can infer exists z: ~Q_4(z). Therefore, the answer is yes.