Consider the following premises. forall z: P(z) -> Q(z). forall z: P(z). exists z: Q(z) <-> S_4(z). Can we infer exists z: Q(z) from them?
 From exists z: Q(z) <-> S_4(z) we can infer exists z: S_4(z) -> Q(z). From forall z: P(z) we can infer forall z: P(z) or S_4(z). Finally, from forall z: P(z) -> Q(z), exists z: S_4(z) -> Q(z), forall z: P(z) or S_4(z) we can infer exists z: Q(z). Therefore, the answer is yes.