Consider the following premises. There is at least one x for which if x were sad, then x is running. For all x, x is not sad. For all x, x is sad or x is not a carpenter. For all x, if x goes running, then x is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: There is at least one x for which x is not a carpenter.
 From the fact that there is at least one x for which if x were sad, then x is running, and that for all x, if x goes running, then x is a carpenter we can infer that there is at least one x for which if x were sad, then x is a carpenter via existential transitivity. From the fact that for all x, x is sad or x is not a carpenter we can infer that for all x, if x were a carpenter, then x is sad via universal rewriting. From the fact that there is at least one x for which if x were sad, then x is a carpenter, and that for all x, if x were a carpenter, then x is sad we can infer that there is at least one x for which x is sad if and only if x is a carpenter via existential biconditional introduction. Finally, from the fact that there is at least one x for which x is sad if and only if x is a carpenter, and that for all x, x is not sad we can infer that there is at least one x for which x is not a carpenter via existential biconditional elimination. Therefore, the answer is yes.