In this letter, we derive another exact bit error rate (BER) for decode-and-forward (DF) relay systems over Rayleigh fading channels. At first, our focus is on fixed-DF (FDF) relay schemes in which the probability density function (PDF) is derived based on error-events at relay nodes. Some insight into how erroneous detection and transmission at relay nodes affect both the combined signal-to-noise ratio (SNR) and the averaged BER is obtained, and cooperative diversity is observed from the closed-form BER expression. In addition, the developed analytical method is extended to adaptive-DF (ADF) schemes and the exact BER expressions are derived. Simulation results are finally presented to validate the analysis.