Consider the following premises. t or u. t -> w. r_4 <-> w. u -> w. Can we infer ~r_4 from them? If possible, name the inference rules being used at each step.
 From t -> w, u -> w, t or u we can infer w via disjunction elimination. From r_4 <-> w we can infer w -> r_4 via biconditional elimination. Finally, from w -> r_4, w we can infer r_4 via modus ponens, which contradicts ~r_4. Therefore, the answer is no.