Consider the following premises. p_3 -> t. t -> q_1. p <-> t. p or p_3. Can we infer ~q_1 from them? If possible, name the inference rules being used at each step.
 From p <-> t we can infer p -> t via biconditional elimination. From p -> t, p_3 -> t, p or p_3 we can infer t via disjunction elimination. Finally, from t -> q_1, t we can infer q_1 via modus ponens, which contradicts ~q_1. Therefore, the answer is no.