So you know what Bell's theorum is abstractly?
The following explantion is based on the one in Quantum Mechanics by Alastair Rae published by the IoP, though I've added a few bits and taken out others and generally changed the wording:
Consider a local hidden variable theory (LHVT) and a system of spin half atoms, as it's a LHVT the result of the measuremnt of the spins will be pre-detirmined before the measuremnt takes place.
Now consider the components of the spin in three directions (1,2,3). a set of N atoms will contain a subset of n(1+,2+,3+) particles, each of them would mean apostive result if there spin was measured in any of the three directions, wheras, particles that are members of the n(1-,2+,3+) would yield a negative result in the 1 direction and a postive in the 2 and 3 directions, etc., therefore the set of N particles conatins 8 mutually exclusive subsets as defined by their spin components in the three directions. You cannot define which atom is in which subset as measuring one spin component would change the others but as this is an LHVT all atoms must belong to one of the subsets.
Now let's say that each atom in the subset N is a member of an entangled pair (though obviously as we've assumed a LHVT they won't be truly entangled but they're measuremnts are still dependent on each other though pre-detirmined). By measuring the spin component of 1 member of the entangled pair in say the 1 direction and the other member in the 2 direction we will be able to know the components of spin in both the 1 and 2 directions of a particle by only disturbing it once as the measurements do not affect each other. we can by measuring these two components create 5 new subsets:
n(1+,2+) = n(1+,2+,3+) + n(1+,2+,3-)
n(1+,3+) = n(1+,2+,3+) + n(1+,2-,3+)
n(2-,3+) = n(1+,2-,3+) + n(1-,2-,3+)
etc.
If N is large enough we should be able to effectively measure any of the above three sets, also from the above we can detirmine:
n(1+,2+) - n(1+,3+) + n(2+,3-) = n(1+,2+,3-) + n(1-,2-,3+)
which means that n(1+,2+) - n(1+,3+) + n(2+,3-) >= 0 (is greaterbthan or equal to zero)
The above is one way of stating Bell's inequality
Putting in the QM predictions for this (where θ12 is the anle between 1 and 2 obtained from the first equation):
n(1+,2+) =NP+-(θ12)
cos2 θ12/2 - cos2 θ13/2 + sin2 θ23/3 >= 0
Now let's say that all three measuremnt directions are in the same plane (as they are allowed to be), therefore:
θ12+θ23 = θ13, now specializing further and conmsidering the case when θ13 = 3θ12 and now taking θ12/2 as θ we get:
cos2θ + sin22θ¸ - cos23θ >= 0
Now we can then plot this function we find that when θ = 20°, among others, the value is actually negative (-0.22) therefore the last equation cannot be true and the predictions of a LHVT differs from those of QM and both cannot be true.
