Consider the following premises. exists x: P(x) <-> T_3(x). forall x: P_3(x) -> P(x). forall x: T_3(x) -> P_3(x). Can we infer forall x: ~(P(x) <-> T_3(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: T_3(x) -> P_3(x), forall x: P_3(x) -> P(x) we can infer forall x: T_3(x) -> P(x) via universal transitivity. From exists x: P(x) <-> T_3(x) we can infer exists x: P(x) -> T_3(x) via existential biconditional elimination. Finally, from exists x: P(x) -> T_3(x), forall x: T_3(x) -> P(x) we can infer exists x: P(x) <-> T_3(x) via existential biconditional introduction, which contradicts forall x: ~(P(x) <-> T_3(x)). Therefore, the answer is no.