Consider the following premises. forall z_2: S_2(z_2) -> U_1(z_2). forall z_2: P(z_2) -> S_2(z_2). exists z_2: U_1(z_2) -> P(z_2). Can we infer forall z_2: ~(S_2(z_2) <-> P(z_2)) from them? If possible, name the inference rules being used at each step.
 From forall z_2: S_2(z_2) -> U_1(z_2), exists z_2: U_1(z_2) -> P(z_2) we can infer exists z_2: S_2(z_2) -> P(z_2) via existential transitivity. Finally, from exists z_2: S_2(z_2) -> P(z_2), forall z_2: P(z_2) -> S_2(z_2) we can infer exists z_2: S_2(z_2) <-> P(z_2) via existential biconditional introduction, which contradicts forall z_2: ~(S_2(z_2) <-> P(z_2)). Therefore, the answer is no.