Consider the following premises. forall x_8: P_8(x_8) -> T(x_8). ~P_8(a_2). forall x_8: T(x_8) -> P_8(x_8). forall x_8: T(x_8) -> P_8(x_8). Can we infer T(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) -> T(x_8), forall x_8: T(x_8) -> P_8(x_8) we can infer forall x_8: P_8(x_8) <-> T(x_8) via universal biconditional introduction. From forall x_8: P_8(x_8) <-> T(x_8) we can infer forall x_8: P_8(x_8) -> T(x_8) via universal biconditional elimination. From forall x_8: T(x_8) -> P_8(x_8), forall x_8: P_8(x_8) -> T(x_8) we can infer forall x_8: T(x_8) <-> P_8(x_8) via universal biconditional introduction. From forall x_8: T(x_8) <-> P_8(x_8) we can infer T(a_2) <-> P_8(a_2) via universal instantiation. Finally, from T(a_2) <-> P_8(a_2), ~P_8(a_2) we can infer ~T(a_2) via biconditional elimination, which contradicts T(a_2). Therefore, the answer is no.