The condition you identify as "IIA" isn't Arrow's condition. IIA in Arrow's theorem is best understood as the principle that only information on the ballot is r

The condition you identify as "IIA" isn't Arrow's condition. IIA in Arrow's theorem is best understood as the principle that only information on the ballot is relevant to the vote. The condition you cite is a much stronger counterfactual property: that if x beats y, then x beats y even if we introduce z. But this is not Arrow's condition (see Bordes and Tideman on this).