Consider the following premises. exists y_2: U_1(y_2). forall y_2: W_1(y_2) or ~T(y_2). forall y_2: U(y_2) -> W_1(y_2). forall y_2: U_1(y_2) -> U(y_2). Can we infer exists y_2: W_1(y_2) from them? If possible, name the inference rules being used at each step.
 From forall y_2: U_1(y_2) -> U(y_2), forall y_2: U(y_2) -> W_1(y_2) we can infer forall y_2: U_1(y_2) -> W_1(y_2) via universal transitivity. From exists y_2: U_1(y_2) we can infer exists y_2: U_1(y_2) or T(y_2) via existential addition. From forall y_2: W_1(y_2) or ~T(y_2) we can infer forall y_2: T(y_2) -> W_1(y_2) via universal rewriting. Finally, from forall y_2: U_1(y_2) -> W_1(y_2), forall y_2: T(y_2) -> W_1(y_2), exists y_2: U_1(y_2) or T(y_2) we can infer exists y_2: W_1(y_2) via existential disjunction elimination. Therefore, the answer is yes.