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