Consider the following premises. ~u. s_3 <-> q_21. u <-> s_3. q_1 -> r. ~q_21 -> r. Can we infer ~r from them? If possible, name the inference rules being used at each step.
 From s_3 <-> q_21 we can infer q_21 -> s_3 via biconditional elimination. From u <-> s_3, ~u we can infer ~s_3 via biconditional elimination. From q_21 -> s_3, ~s_3 we can infer ~q_21 via modus tollens. From ~q_21 we can infer ~q_21 or q_1 via addition. Finally, from ~q_21 -> r, q_1 -> r, ~q_21 or q_1 we can infer r via disjunction elimination, which contradicts ~r. Therefore, the answer is no.