During the past decade, we learned that the structure of contact networks plays a crucial role in the spread of diseases. Most theoretical studies addressing this issue assume that contact networks are static entities, whereas the actual disease paths continuously reshape based on local social dynamics. This work aims to achieve a better understanding of disease spreading in populations characterized by a dynamically structured contact network where contacts appear and disappear over time. The network dynamics are entangled with the disease dynamics, as individuals may have access to local information that makes them aware of both the existence of the disease and the health status of their contacts, allowing them to minimize exposure to infection. Here we show the equivalence between disease propagation in an adaptive contact network and that in a well-mixed population with a rescaled transmission probability, which depends also on the fraction of infected in the population. Thus, one can emulate the effect of an adaptive contact network with a simple correction of the transmission probability. This result is obtained in the limit where network adaptation proceeds much faster than disease spreading, but we demonstrate that it also holds for a much wider range of scenarios.