Consider the following premises. forall x_10: P_10(x_10) -> U_2(x_10). forall x_10: U_2(x_10) -> P_10(x_10). P_10(g_2) -> r. Can we infer U_2(g_2) -> r from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) -> U_2(x_10), forall x_10: U_2(x_10) -> P_10(x_10) we can infer forall x_10: P_10(x_10) <-> U_2(x_10) via universal biconditional introduction. From forall x_10: P_10(x_10) <-> U_2(x_10) we can infer forall x_10: U_2(x_10) -> P_10(x_10) via universal biconditional elimination. From forall x_10: U_2(x_10) -> P_10(x_10) we can infer U_2(g_2) -> P_10(g_2) via universal instantiation. Finally, from U_2(g_2) -> P_10(g_2), P_10(g_2) -> r we can infer U_2(g_2) -> r via transitivity. Therefore, the answer is yes.