Consider the following premises. ~t. p_15 <-> t. p_1 -> p_15. ~p_1 -> q. p_15 -> p_1. Can we infer ~q from them? If possible, name the inference rules being used at each step.
 From p_15 <-> t, ~t we can infer ~p_15 via biconditional elimination. From p_15 -> p_1, p_1 -> p_15 we can infer p_15 <-> p_1 via biconditional introduction. From p_15 <-> p_1, ~p_15 we can infer ~p_1 via biconditional elimination. Finally, from ~p_1 -> q, ~p_1 we can infer q via modus ponens, which contradicts ~q. Therefore, the answer is no.