Consider the following premises. t_3 <-> P_1(a_20). t_3 -> p. forall y: P(y) -> P_1(y). p -> P(a_20). Can we infer P(a_20) <-> P_1(a_20) from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) -> P_1(y) we can infer P(a_20) -> P_1(a_20) via universal instantiation. From t_3 -> p, p -> P(a_20) we can infer t_3 -> P(a_20) via transitivity. From t_3 <-> P_1(a_20) we can infer P_1(a_20) -> t_3 via biconditional elimination. From P_1(a_20) -> t_3, t_3 -> P(a_20) we can infer P_1(a_20) -> P(a_20) via transitivity. Finally, from P(a_20) -> P_1(a_20), P_1(a_20) -> P(a_20) we can infer P(a_20) <-> P_1(a_20) via biconditional introduction. Therefore, the answer is yes.