Consider the following premises. forall z: S_3(z) -> P(z). forall z: P(z) -> S_3(z). forall z: P(z) -> S_3(z). exists z: ~S_3(z). Can we infer forall z: P(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P(z) -> S_3(z), forall z: S_3(z) -> P(z) we can infer forall z: P(z) <-> S_3(z) via universal biconditional introduction. From forall z: P(z) <-> S_3(z) we can infer forall z: S_3(z) -> P(z) via universal biconditional elimination. From forall z: P(z) -> S_3(z), forall z: S_3(z) -> P(z) we can infer forall z: P(z) <-> S_3(z) via universal biconditional introduction. Finally, from forall z: P(z) <-> S_3(z), exists z: ~S_3(z) we can infer exists z: ~P(z) via existential biconditional elimination, which contradicts forall z: P(z). Therefore, the answer is no.