Consider the following premises. forall x: P(x) <-> P_2(x). P(h) -> t. Can we infer ~(P_2(h) -> t) from them? If possible, name the inference rules being used at each step.
 From forall x: P(x) <-> P_2(x) we can infer P(h) <-> P_2(h) via universal instantiation. From P(h) <-> P_2(h) we can infer P_2(h) -> P(h) via biconditional elimination. Finally, from P_2(h) -> P(h), P(h) -> t we can infer P_2(h) -> t via transitivity, which contradicts ~(P_2(h) -> t). Therefore, the answer is no.