Consider the following premises. u -> t. t -> t_1. t or ~t_1. u or t_1. Can we infer ~t from them? If possible, name the inference rules being used at each step.
 From t or ~t_1 we can infer t_1 -> t via rewriting. From t_1 -> t, t -> t_1 we can infer t_1 <-> t via biconditional introduction. From t_1 <-> t we can infer t_1 -> t via biconditional elimination. Finally, from u -> t, t_1 -> t, u or t_1 we can infer t via disjunction elimination, which contradicts ~t. Therefore, the answer is no.