Consider the following premises. forall y_1: U_1(y_1) <-> P_3(y_1). exists y_1: P_3(y_1) <-> U_1(y_1). Can we infer exists y_1: P_3(y_1) -> U_1(y_1) from them? If possible, name the inference rules being used at each step.
 From exists y_1: P_3(y_1) <-> U_1(y_1) we can infer exists y_1: U_1(y_1) -> P_3(y_1) via existential biconditional elimination. From forall y_1: U_1(y_1) <-> P_3(y_1) we can infer forall y_1: P_3(y_1) -> U_1(y_1) via universal biconditional elimination. From exists y_1: U_1(y_1) -> P_3(y_1), forall y_1: P_3(y_1) -> U_1(y_1) we can infer exists y_1: U_1(y_1) <-> P_3(y_1) via existential biconditional introduction. Finally, from exists y_1: U_1(y_1) <-> P_3(y_1) we can infer exists y_1: P_3(y_1) -> U_1(y_1) via existential biconditional elimination. Therefore, the answer is yes.