Consider the following premises. forall z: P(z) -> T(z). forall z: ~W_1(z). exists z: T(z) <-> W_1(z). Can we infer exists z: ~P(z) from them? If possible, name the inference rules being used at each step.
 From exists z: T(z) <-> W_1(z) we can infer exists z: T(z) -> W_1(z) via existential biconditional elimination. From forall z: P(z) -> T(z), exists z: T(z) -> W_1(z) we can infer exists z: P(z) -> W_1(z) via existential transitivity. Finally, from exists z: P(z) -> W_1(z), forall z: ~W_1(z) we can infer exists z: ~P(z) via existential modus tollens. Therefore, the answer is yes.