You need this sort of axiom to eliminate superdeterminism. If an experimenter's choice of direction is fully determined by their past light-cone then a shared physical state in the overlapping past light-cones of A and B could conceivably drum up any correlations it wants.
It's a conspiratorial line of thought on a par with Descartes' evil demon, but in a rigorous proof you have to rule these sorts of things out.
The axiom comes into play at the top-right of p.229:
> ...since by MIN the response of b cannot vary with x, y, z, θ^G_0 is a function just of w
Right, I see how it's being used in the proof, my point is that what the theorem proves, regardless of how the notion of "is determined by" is clarified, is not at all interesting or surprising.
It's essentially a recasting of Bell's Theorem; in that sense it's both interesting and surprising if you thought that local hidden variables might still explain quantum theory. All you need - for both this and Bell's proof - is that directions are chosen with sufficient "randomness" or "freedom" so as to be considered independent of their past light cones. At least, independent enough so that the previously quoted step is possible.
It's a conspiratorial line of thought on a par with Descartes' evil demon, but in a rigorous proof you have to rule these sorts of things out.
The axiom comes into play at the top-right of p.229:
> ...since by MIN the response of b cannot vary with x, y, z, θ^G_0 is a function just of w