Consider the following premises. exists y_1: U_2(y_1). forall y_1: W(y_1) or U(y_1). forall y_1: U(y_1) <-> P_2(y_1). forall y_1: W(y_1) -> P_2(y_1). Can we infer exists y_1: P_2(y_1) and U_2(y_1) from them? If possible, name the inference rules being used at each step.
 From forall y_1: U(y_1) <-> P_2(y_1) we can infer forall y_1: U(y_1) -> P_2(y_1) via universal biconditional elimination. From forall y_1: W(y_1) -> P_2(y_1), forall y_1: U(y_1) -> P_2(y_1), forall y_1: W(y_1) or U(y_1) we can infer forall y_1: P_2(y_1) via universal disjunction elimination. Finally, from forall y_1: P_2(y_1), exists y_1: U_2(y_1) we can infer exists y_1: P_2(y_1) and U_2(y_1) via existential conjunction. Therefore, the answer is yes.