Consider the following premises. exists y: R_4(y) <-> P(y). forall y: ~P(y). Can we infer exists y: ~R_4(y) from them? If possible, name the inference rules being used at each step.
 From exists y: R_4(y) <-> P(y) we can infer exists y: R_4(y) -> P(y) via existential biconditional elimination. Finally, from exists y: R_4(y) -> P(y), forall y: ~P(y) we can infer exists y: ~R_4(y) via existential modus tollens. Therefore, the answer is yes.