Consider the following premises. r <-> T(a_7). r <-> u. ~u. forall z: P_1(z) <-> T(z). Can we infer ~P_1(a_7) from them? If possible, name the inference rules being used at each step.
 From r <-> u, ~u we can infer ~r via biconditional elimination. From forall z: P_1(z) <-> T(z) we can infer P_1(a_7) <-> T(a_7) via universal instantiation. From r <-> T(a_7), ~r we can infer ~T(a_7) via biconditional elimination. Finally, from P_1(a_7) <-> T(a_7), ~T(a_7) we can infer ~P_1(a_7) via biconditional elimination. Therefore, the answer is yes.