I think metaman's reconstruction is correct. The proof is rather messy if we unpack it, and it also reveals some nasty stuff that pfbradley sweeps under the carpett.
IMO the argument is highly suspect. It is a pretty simple case of begging the question.
1) Either materialism, idealism or dualism is the case. (exhaustive disjunction of possible ontologies of the world)
2) If materialism is the case, the mental is reducible to the physical& some physical substance exists. (minimal materialism)
3) If dualism is true, then the mental is not reducible to the physical & some mental & some physical exists. (minimal dualism)
4) If idealism is true, then the physical is reducible to the mental & some mental substance exists. (minimal idealism)
5) the mental is not reducible to the physical (Begging the question!)
6) it is not the case that some mental & some physical exists. (Begging the question)
7Assumption: idealism is false. (assumption used in deriving contradiction from 1)

Idealism (first disjunct of 1)
9) kontradiction from 7&8
10) materialism is false from 5&2. (via tollens)
11) materialism from 1.
12) kontradiction from 10&11
13) dualism is false from 3 &6 (via tollens)
14) dualism
15) kontradiction
16) kontradiction via disjunction elimination
17) double negate idealism
Hence: idealism is true.
This proof works solely because of the assumption that both dualism and materialism is false + the assumption that idealism is the only living position of the disjunction in 1. Apart from the fact that both the dualist and materialist position is given an unfair formulation, the argument is simply not persuasive. Of course you could try another type of argument relying on, say, simplicity. But then you have a weird account of simplicity, since idealism gains simplicity by killing of a lot of stuff that should have a spot in our theory of the world, for instance other minds. Also, materialism is a living contender, and new breeds of nonreductive materialism have gained popularity. Also, dualism is having a revival in western philosophy. A lot of exciting research is going on. The only position with no followers is, fyi, idealism.
Could you try giving some backing to the question begging premisses? For instance explain why we ought to think all types of materialism out of the game. Also, how about adressing the issues with idealism? It has at least as great problems as the other two. You are being unfair imo. By your standards we ought to rule out the disjunction of idealism, dualism and materialism and conclude the world has no ontology. Or perhaps we should admit that, at present, we can't rule out any of the positions. But given the status quo, I opt for some kind of softcore physicalism. Dualism has its charm, and contemporary philosophy is starting to face up the the problems of reductionism.