Consider the following premises. r_1 -> p_1. p_1 -> r_1. r_1 -> ~r. r. Can we infer ~p_1 from them? If possible, name the inference rules being used at each step.
 From r_1 -> p_1, p_1 -> r_1 we can infer r_1 <-> p_1 via biconditional introduction. From r_1 -> ~r, r we can infer ~r_1 via modus tollens. Finally, from r_1 <-> p_1, ~r_1 we can infer ~p_1 via biconditional elimination. Therefore, the answer is yes.