Electricity transmission expansion has suffered many delays in Europe in recent decades, despite its importance for integrating renewable electricity into the energy system. A hydrogen network which reuses the existing fossil gas network would not only help supply demand for low-emission fuels, but could also help to balance variations in wind and solar energy across the continent and thus avoid power grid expansion. We pursue this idea by varying the allowed expansion of electricity and hydrogen grids in net-zero CO2 scenarios for a sector-coupled European energy system with high shares of renewables and self-sufficient supply.