quantum mechanics - Assumptions in Bell's Theorem

It is often Stated that Bell's Theorem is equivalent to the statement: No theory of Local Hidden Variables can reproduce all of the predictions of quantum mechanics. I see nowhere in Bell's Theorem the assumption of hidden variables. As far as I can tell, there is one assumption: locality, which is shown to conflict with quantum mechanics. If I am wrong on this, could someone demonstrate exactly how Bell failed to demonstrate that quantum mechanics is non-local by showing how the assumption of hidden variables is used in his theorem? To me -- and by the way Bell -- the problem is locality.