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The universality for the eigenvalue spacing statistics of generalized Wignermatrices was established in our previous work \cite{EYY} under certainconditions on the probability distributions of the matrix elements. A majorclass of probability measures excluded in \cite{EYY} are the Bernoullimeasures. In this paper, we extend the universality result of \cite{EYY} toinclude the Bernoulli measures so that the only restrictions on the probabilitydistributions of the matrix elements are the subexponential decay and thenormalization condition that the variances in each row sum up to one. The newingredient is a strong local semicircle law which improves the error estimateon the Stieltjes transform of the empirical measure of the eigenvalues from theorder $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary partof the spectral parameter in the definition of the Stieltjes transform and $N$is the size of the matrix.

Universality for generalized Wigner matrices with Bernoulli distribution