Consider the following premises. forall z: R(z) -> P(z). exists z: P(z) -> T_4(z). forall z: P(z) -> R(z). forall z: T_4(z) <-> P(z). forall z: ~R(z). Can we infer forall z: T_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P(z) -> R(z), forall z: R(z) -> P(z) we can infer forall z: P(z) <-> R(z) via universal biconditional introduction. From forall z: T_4(z) <-> P(z) we can infer forall z: T_4(z) -> P(z) via universal biconditional elimination. From forall z: P(z) <-> R(z), forall z: ~R(z) we can infer forall z: ~P(z) via universal biconditional elimination. From exists z: P(z) -> T_4(z), forall z: T_4(z) -> P(z) we can infer exists z: P(z) <-> T_4(z) via existential biconditional introduction. Finally, from exists z: P(z) <-> T_4(z), forall z: ~P(z) we can infer exists z: ~T_4(z) via existential biconditional elimination, which contradicts forall z: T_4(z). Therefore, the answer is no.