Consider the following premises. forall x_8: T_4(x_8) or T(x_8). forall x_8: T_4(x_8) <-> P_1(x_8). exists x_8: P_1(x_8) or ~T(x_8). Can we infer forall x_8: ~P_1(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: T_4(x_8) <-> P_1(x_8) we can infer forall x_8: T_4(x_8) -> P_1(x_8) via universal biconditional elimination. From exists x_8: P_1(x_8) or ~T(x_8) we can infer exists x_8: T(x_8) -> P_1(x_8) via existential rewriting. Finally, from forall x_8: T_4(x_8) -> P_1(x_8), exists x_8: T(x_8) -> P_1(x_8), forall x_8: T_4(x_8) or T(x_8) we can infer exists x_8: P_1(x_8) via existential disjunction elimination, which contradicts forall x_8: ~P_1(x_8). Therefore, the answer is no.