Despite many policy instruments to mitigate climate change, literature on distributional justice has mainly focused on carbon taxes. We focus on two regulatory instruments that could speed up climate change mitigation in Switzerland: banning the use of fossil fuel cars and boilers in the residential sector. By merging data from Swiss Household Budget Survey, Mobility and Transport Microcensus, and Swiss Household Energy Demand Survey, we quantify the mitigation potential and distributional justice of various ban designs. We find that, in general, the bans that target more emissions are more egalitarian (i.e. achieve a more equal distribution of vulnerabilities), but less just under the utilitarian, Rawlsian difference, and sufficientarian principles. Under these three principles, the best compromise between mitigation potential and justice is to include exemptions for the lowest income households. An alternative could be more ambitious bans to maximize mitigation potential while compensating for the most vulnerable households. For selected ban designs, we identify which household groups would be most vulnerable and retrieve specific cases of hardship.