The spin Seebeck effect (SSE) consists in the generation of a spin current by a temperature gradient applied in a magnetic film. The SSE is usually detected by an electric voltage generated in a metallic layer in contact with the magnetic film resulting from the conversion of the spin current into charge current by means of the inverse spin Hall effect. The SSE has been widely studied in bilayers made of the insulating ferrimagnet yttrium iron garnet (YIG) and metals with large spin-orbit coupling such as platinum. Recently the SSE has been observed in bilayers made of the antiferromagnet $\mathrm{Mn}{\mathrm{F}}_{2}$ and Pt, revealing dependences of the SSE voltage on temperature and field very different from the ones observed in YIG/Pt. Here we present a theory for the SSE in structures with an antiferromagnetic insulator (AFI) in contact with a normal metal (NM) that relies on the bulk magnon spin current created by the temperature gradient across the thickness of the AFI/NM bilayer. The theory explains quite well the measured dependences of the SSE voltage on the sample temperature and on the applied magnetic field in $\mathrm{Mn}{\mathrm{F}}_{2}/\mathrm{Pt}$.