Consider the following premises. q_4 <-> r. p_3 -> r. ~q_4. p_3 -> q_1. q_1 -> p_3. Can we infer ~q_1 from them? If possible, name the inference rules being used at each step.
 From q_4 <-> r, ~q_4 we can infer ~r via biconditional elimination. From p_3 -> r, ~r we can infer ~p_3 via modus tollens. From p_3 -> q_1, q_1 -> p_3 we can infer p_3 <-> q_1 via biconditional introduction. Finally, from p_3 <-> q_1, ~p_3 we can infer ~q_1 via biconditional elimination. Therefore, the answer is yes.