A number of cities have embraced building electrification and require all-electric equipment in newly constructed buildings. This paper builds a model of the dynamic process by which electrification in newly constructed buildings spreads to electrification in existing buildings. The key channel of spillover is the rising prices of piped gas resulting from a high share of fixed cost in operating a gas distribution system and a slowly declining customer base. The paper takes into account that there are heterogeneous fixed costs in electrifying existing buildings across households. We show that the resulting dynamics depend crucially on how building owners form expectations about the future price of natural gas. We model near rational expectations formation processes based on level-k thinking and show that the rising price of gas can substantially accelerate the rate of decline of gas throughput, aka speed up electrification of the full building stock. We calibrate the model with micro level data from the ResStock and ComStock databases provided by NREL and illustrate the resulting dynamics based on a random sample with detailed building level information in Washington DC.