Consider the following premises. forall z_1: P(z_1) -> S(z_1). forall z_1: P(z_1) <-> S(z_1). forall z_1: S(z_1) -> P(z_1). Can we infer exists z_1: ~(P(z_1) <-> S(z_1)) from them? If possible, name the inference rules being used at each step.
 From forall z_1: P(z_1) -> S(z_1), forall z_1: S(z_1) -> P(z_1) we can infer forall z_1: P(z_1) <-> S(z_1) via universal biconditional introduction. From forall z_1: P(z_1) <-> S(z_1) we can infer forall z_1: S(z_1) -> P(z_1) via universal biconditional elimination. From forall z_1: P(z_1) <-> S(z_1) we can infer forall z_1: P(z_1) -> S(z_1) via universal biconditional elimination. Finally, from forall z_1: P(z_1) -> S(z_1), forall z_1: S(z_1) -> P(z_1) we can infer forall z_1: P(z_1) <-> S(z_1) via universal biconditional introduction, which contradicts exists z_1: ~(P(z_1) <-> S(z_1)). Therefore, the answer is no.