Consider the following premises. ~U_1(a_12). forall y_1: U_1(y_1) -> P(y_1). U_1(a_12) <-> P(a_12). Can we infer P(a_12) from them? If possible, name the inference rules being used at each step.
 From forall y_1: U_1(y_1) -> P(y_1) we can infer U_1(a_12) -> P(a_12) via universal instantiation. From U_1(a_12) <-> P(a_12) we can infer P(a_12) -> U_1(a_12) via biconditional elimination. From U_1(a_12) -> P(a_12), P(a_12) -> U_1(a_12) we can infer U_1(a_12) <-> P(a_12) via biconditional introduction. Finally, from U_1(a_12) <-> P(a_12), ~U_1(a_12) we can infer ~P(a_12) via biconditional elimination, which contradicts P(a_12). Therefore, the answer is no.