Consider the following premises. forall y_4: T_1(y_4). forall y_4: U_2(y_4). forall y_4: T_1(y_4) -> W_1(y_4). Can we infer exists z: U_2(z) from them? If possible, name the inference rules being used at each step.
 From forall y_4: T_1(y_4) -> W_1(y_4), forall y_4: T_1(y_4) we can infer forall y_4: W_1(y_4) via universal modus ponens. From forall y_4: U_2(y_4), forall y_4: W_1(y_4) we can infer forall y_4: U_2(y_4) and W_1(y_4) via universal conjunction. From forall y_4: U_2(y_4) and W_1(y_4) we can infer forall y_4: U_2(y_4) via universal simplification. From forall y_4: U_2(y_4) we can infer U_2(f_1) via universal instantiation. Finally, from U_2(f_1) we can infer exists z: U_2(z) via existential generalization. Therefore, the answer is yes.