Consider the following premises. For all z, if z plays a game, then z is an artist. For all z, z is a sailor if and only if z is an artist. For all z, z will play a game or z is not a sailor. Can we infer the following from them? If we can, name the inference rule being used: There is at least one z for which if is not true that z will play a game if and only if z is a sailor.
 From the fact that for all z, z will play a game or z is not a sailor we can infer that for all z, if z were a sailor, then z is playing a game via universal rewriting. From the fact that for all z, z is a sailor if and only if z is an artist we can infer that for all z, if z were an artist, then z is a sailor via universal biconditional elimination. From the fact that for all z, if z plays a game, then z is an artist, and that for all z, if z were an artist, then z is a sailor we can infer that for all z, if z plays a game, then z is a sailor via universal transitivity. Finally, from the fact that for all z, if z plays a game, then z is a sailor, and that for all z, if z were a sailor, then z is playing a game we can infer that for all z, z will play a game if and only if z is a sailor via universal biconditional introduction, which contradicts that There is at least one z for which if is not true that z will play a game if and only if z is a sailor. Therefore, the answer is no.