As far as I know, every argument for utility assumes (or implies) that whenever you make an observation, you stop caring about the possible worlds where that observation went differently.
Are you just referring to the VNM theorems or are there other theorems you have in mind?
Note for self: It seems like the independence condition breaks for counterfactual mugging assuming you think we should pay. Assume P is paying $50 and N is not paying, M is receiving $1 million if you would have paid in the counterfactual and zero otherwise. We have N>P but 0.5P+0.5M>0.5N+0.5M in contradiction to independence. The issue is that the value of M is not independent of the choice between P and N.
Are you just referring to the VNM theorems or are there other theorems you have in mind?
Note for self: It seems like the independence condition breaks for counterfactual mugging assuming you think we should pay. Assume P is paying $50 and N is not paying, M is receiving $1 million if you would have paid in the counterfactual and zero otherwise. We have N>P but 0.5P+0.5M>0.5N+0.5M in contradiction to independence. The issue is that the value of M is not independent of the choice between P and N.