Consider the following premises. exists z_4: P(z_4) <-> T(z_4). forall z_4: ~U_2(z_4). forall z_4: P(z_4) <-> U_2(z_4). Can we infer exists z_4: ~T(z_4) from them? If possible, name the inference rules being used at each step.
 From forall z_4: P(z_4) <-> U_2(z_4) we can infer forall z_4: P(z_4) -> U_2(z_4) via universal biconditional elimination. From forall z_4: P(z_4) -> U_2(z_4), forall z_4: ~U_2(z_4) we can infer forall z_4: ~P(z_4) via universal modus tollens. Finally, from exists z_4: P(z_4) <-> T(z_4), forall z_4: ~P(z_4) we can infer exists z_4: ~T(z_4) via existential biconditional elimination. Therefore, the answer is yes.