  The universality for the eigenvalue spacing statistics of generalized Wigner
matrices was established in our previous work \cite{EYY} under certain
conditions on the probability distributions of the matrix elements. A major
class of probability measures excluded in \cite{EYY} are the Bernoulli
measures. In this paper, we extend the universality result of \cite{EYY} to
include the Bernoulli measures so that the only restrictions on the probability
distributions of the matrix elements are the subexponential decay and the
normalization condition that the variances in each row sum up to one. The new
ingredient is a strong local semicircle law which improves the error estimate
on the Stieltjes transform of the empirical measure of the eigenvalues from the
order $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary part
of the spectral parameter in the definition of the Stieltjes transform and $N$
is the size of the matrix.
