  In this Letter, we explore the possibility of developing Bell inequalities
predicated on models using a single Local Hidden Variable (s-LHV), a strict
subset of general LHV models. Because of the less strenuous constraints imposed
by s-LHV models, we were able to derive a contingent Bell inequality in analogy
to the CHSH inequality, but which does not require bounding of measurement
statistics. Following this, we show by explicit example that there are cases of
states that rule out s-LHV models by violating our inequality, but which
nonetheless have a multivariate LHV model. Even so, we show how merely ruling
out s-LHV models is still sufficient to allow for fully device independent
quantum key distribution (QKD) and entanglement witnessing. This being the
case, our inequality illustrates two things. First, it makes fully
device-independent QKD on continuous variables substantially more
straightforward. Second, it shows how the degree of correlation needed to
demonstrate device-independent QKD is distinct from both Bell-nonlocality and
EPR-steering. Although nonlocality in general requires ruling out all LHV
models, s-LHV models are sufficiently useful to warrant further exploration.
