Consider the following premises. forall x_2: ~U_2(x_2). forall x_2: P_1(x_2) <-> U_2(x_2). P_1(b) <-> q_4. Can we infer ~q_4 from them? If possible, name the inference rules being used at each step.
 From forall x_2: P_1(x_2) <-> U_2(x_2), forall x_2: ~U_2(x_2) we can infer forall x_2: ~P_1(x_2) via universal biconditional elimination. From forall x_2: ~P_1(x_2) we can infer ~P_1(b) via universal instantiation. From P_1(b) <-> q_4 we can infer q_4 -> P_1(b) via biconditional elimination. Finally, from q_4 -> P_1(b), ~P_1(b) we can infer ~q_4 via modus tollens. Therefore, the answer is yes.