  Hardy-type arguments manifest Bell nonlocality in one of the simplest
possible ways. Except for demonstrating non-classical signature of entangled
states in question, they can also serve for device independent self-testing of
states, as shown e.g. in [Phys. Rev. Lett. 109, 180401 (2012)]. Here we develop
and broaden these results to an extended version of the Hardy's argument, often
referred as Cabello's nonlocality argument. We show that alike in the simpler
case of Hardy's nonlocality argument, the maximum quantum value for Cabello's
nonlocality is achieved by a pure two qubit state and projective measurements
that are unique up to local isometries. We also examine the properties of a
more realistic case when small errors in the ideal constraints are accepted
within the probabilities obtained and prove that also in this case two qubit
state and measurements are sufficient for obtaining the maximum quantum
violation of the classical bound.
