Consider the following premises. For all x_25, x_25 is a poet. For all x_25, x_25 will not listen to a song. There is at least one x_25 for which if x_25 were a poet, then x_25 is a janitor. For all x_25, if x_25 were a lawyer, then x_25 is not a sailor. For all x_25, x_25 is a janitor if and only if x_25 will listen to a song. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x_25 for which x_25 is not a lawyer.
 From the fact that for all x_25, x_25 is a janitor if and only if x_25 will listen to a song, and that for all x_25, x_25 will not listen to a song we can infer that for all x_25, x_25 is not a janitor via universal biconditional elimination. From the fact that there is at least one x_25 for which if x_25 were a poet, then x_25 is a janitor, and that for all x_25, x_25 is a poet we can infer that there is at least one x_25 for which x_25 is a janitor via existential modus ponens. From the fact that there is at least one x_25 for which x_25 is a janitor we can infer that there is at least one x_25 for which x_25 is a janitor or x_25 is a sailor via existential addition. From the fact that there is at least one x_25 for which x_25 is a janitor or x_25 is a sailor, and that for all x_25, x_25 is not a janitor we can infer that there is at least one x_25 for which x_25 is a sailor via existential disjunctive syllogism. Finally, from the fact that for all x_25, if x_25 were a lawyer, then x_25 is not a sailor, and that there is at least one x_25 for which x_25 is a sailor we can infer that there is at least one x_25 for which x_25 is not a lawyer via existential modus tollens. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.