Consider the following premises. u -> T(i_4). forall y: P(y) <-> T(y). t -> u. Can we infer ~(t -> P(i_4)) from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) <-> T(y) we can infer P(i_4) <-> T(i_4) via universal instantiation. From t -> u, u -> T(i_4) we can infer t -> T(i_4) via transitivity. From P(i_4) <-> T(i_4) we can infer T(i_4) -> P(i_4) via biconditional elimination. Finally, from t -> T(i_4), T(i_4) -> P(i_4) we can infer t -> P(i_4) via transitivity, which contradicts ~(t -> P(i_4)). Therefore, the answer is no.