Consider the following premises. q <-> q_4. q_2 or q. q_4 or ~q_2. Can we infer ~q_4 from them? If possible, name the inference rules being used at each step.
 From q <-> q_4 we can infer q -> q_4 via biconditional elimination. From q_4 or ~q_2 we can infer q_2 -> q_4 via rewriting. Finally, from q_2 -> q_4, q -> q_4, q_2 or q we can infer q_4 via disjunction elimination, which contradicts ~q_4. Therefore, the answer is no.