Consider the following premises. forall x: T_3(x) <-> P(x). forall x: T_3(x) <-> P(x). Can we infer exists 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(x) we can infer forall x: T_3(x) -> P(x) via universal biconditional elimination. From forall x: T_3(x) <-> P(x) we can infer forall x: P(x) -> T_3(x) via universal biconditional elimination. Finally, from forall x: P(x) -> T_3(x), forall x: T_3(x) -> P(x) we can infer forall x: P(x) <-> T_3(x) via universal biconditional introduction, which contradicts exists x: ~(P(x) <-> T_3(x)). Therefore, the answer is no.