Consider the following premises. q or q_3. w or ~q_3. q_1 -> w. q -> q_1. Can we infer ~w from them? If possible, name the inference rules being used at each step.
 From w or ~q_3 we can infer q_3 -> w via rewriting. From q -> q_1, q_1 -> w we can infer q -> w via transitivity. Finally, from q -> w, q_3 -> w, q or q_3 we can infer w via disjunction elimination, which contradicts ~w. Therefore, the answer is no.