Is it possible to have one quest start by some event triggered in another quest?

Ok... so I have the Wrath of Sithis that spawns when the player sleeps only when the quest for it to appear is entered into the console.
I need to have the quest that spawns the wrath be started when the player is evicted from the Dark Brotherhood. So what I'm trying to do is implement a startquest property into the DBEviction quest, but that just isn't working. I'm probably doing it wrong... which begs the question, is this even possible? If so, how do I make it happen?