I upvoted this and I think you proved SIA in a very clever way, but I still don't quite understand why SIA counters the Doomsday argument.

Imagine two universes identical to our own up to the present day. One universe is destined to end in 2010 after a hundred billion humans have existed, the other in 3010 after a hundred trillion humans have existed. I agree that knowing nothing, we would expect a random observer to have a thousand times greater chance of living in the long-lasting universe.

But given that we know this particular random observer is alive in 2009, I would think there's an equal chance of them being in both universes, because both universes contain an equal number of people living in 2009. So my knowledge that I'm living in 2009 screens off any information I should be able to get from the SIA about whether the universe ends in 2010 or 3010. Why can you still use the SIA to prevent Doomsday?

[analogy: you have two sets of numbered balls. One is green and numbered from 1 to 10. The other is red and numbered from 1 to 1000. Both sets are mixed together. What's the probability a randomly chosen ball is red? 1000/1010. Now I tell you the ball has number "6" on it. What's the probability it's red? 1/2. In this case, Doomsday argument still applies (any red or green ball will correctly give information about the number of red or green balls) but SIA doesn't (any red or green ball, given that it's a number shared by both red and green, gives no information on whether red or green is larger.)]