A channel coding achievability bound expressed in terms of the ratio between two Neyman-Pearson β functions is proposed. This bound is the dual of a converse bound established earlier by Polyanskiy and Verdú (2014). The new bound turns out to simplify considerably the analysis in situations where the channel output distribution is not a product distribution, for example due to a cost constraint or a structural constraint (such as orthogonality or constant composition) on the channel inputs. Connections to existing bounds in the literature are discussed. The bound is then used to derive 1) the channel dispersion of additive non-Gaussian noise channels with random Gaussian codebooks, 2) the channel dispersion of an exponential-noise channel, 3) a second-order expansion for the minimum energy per bit of an additive white Gaussian noise channel, and 4) a lower bound on the maximum coding rate of a multiple-input multiple-output Rayleigh-fading channel with perfect channel state information at the receiver, which is the tightest known achievability result.