Consider the following premises. ~p_1. r_4 -> q_6. p_1 <-> q_6. q_6 -> r_4. Can we infer r_4 from them? If possible, name the inference rules being used at each step.
 From r_4 -> q_6, q_6 -> r_4 we can infer r_4 <-> q_6 via biconditional introduction. From p_1 <-> q_6, ~p_1 we can infer ~q_6 via biconditional elimination. Finally, from r_4 <-> q_6, ~q_6 we can infer ~r_4 via biconditional elimination, which contradicts r_4. Therefore, the answer is no.