It will work. The trigger only runs for the player(s) who have the name IceBaby. This will cause them to drop from the other players and, of course, be defeated (since that's what your trigger does).

If two people had the name IceBaby, they would not necessarily drop from each other, but they would drop from all players that do not have the name IceBaby. Both players with the name IceBaby would also be defeated (since that's what your trigger does).

Thank u! Works great!