Consider the following premises. forall z_4: Q(z_4) -> P(z_4). forall z_4: Q(z_4) or ~S(z_4). forall z_4: P(z_4) -> S(z_4). Can we infer exists z_4: ~(S(z_4) <-> P(z_4)) from them?
 From forall z_4: Q(z_4) or ~S(z_4) we can infer forall z_4: S(z_4) -> Q(z_4). From forall z_4: S(z_4) -> Q(z_4), forall z_4: Q(z_4) -> P(z_4) we can infer forall z_4: S(z_4) -> P(z_4). Finally, from forall z_4: S(z_4) -> P(z_4), forall z_4: P(z_4) -> S(z_4) we can infer forall z_4: S(z_4) <-> P(z_4), which contradicts exists z_4: ~(S(z_4) <-> P(z_4)). Therefore, the answer is no.