To search new hard or superhard materials in transition-metal light-element compounds is a current research focus. Most of the past researches focused on binary phases such as transition metal borides, carbides and nitrides, while the researches on ternary phases were relatively rare. The single crystals Nb₃B₃C and Nb₄B₃C₂ were synthesized recently by using Al-Cu alloys as auxiliary metals and their structures were determined by Hillebrechtand Gebhardt. In the present paper, 29 TM₃B₃C and 29 TM₄B₃C₂ configurations are constructed by TM atoms (TM = Sc to Zn, Y to Cd, Hf to Hg) replacing Nb atoms in the known Nb₃B₃C and Nb₄B₃C₂ configuration. By calculating the formation energy from first-principles density functional theories, only 13 TM₃B₃C and 11 TM₄B₃C₂ phases are stable compared with the three elemental substances of TM, boron and carbon. However compared with the most competing phases, only Ta₃B₃C, Nb₃B₃C and Nb₄B₃C₂ phases are stable thermodynamically. The metastable Ta₄B₃C₂ phase at 0 K becomes stable when temperature is higher than 250 K. Thus two new phases of Ta₃B₃C and Ta₄B₃C₂ are uncovered to be stable thermodynamically. Global structure searches conducted by the popular USPEX and CALYPSO softwares prove the Ta₃B₃C and Ta₄B₃C₂ phases to be the most favorable energetically. By calculating the phonon dispersion curves of the Ta₃B₃C and Ta₄B₃C₂ phase, no imaginary phonon frequencies are observed in the whole Brillouin zone, which demonstrates the dynamical stability of the Ta₃B₃C and Ta₄B₃C₂ phase. The calculated elastic constant of the Ta₃B₃C and Ta₄B₃C₂ phases satisfy the criteria of mechanical stability, showing that the Ta₃B₃C and Ta₄B₃C₂ phase are stable mechanically. The calculations of band structure and density of state show that the Ta₃B₃C and Ta₄B₃C₂ phases are both conducting, which mainly arises from the d electrons of Ta atoms. The calculated bulk modulus and shear modulus of the Ta₃B₃C and Ta₄B₃C₂ phases show their brittle nature. The hardness of Ta₃B₃C and Ta₄B₃C₂ phase are nearly the same and calculated to be about 26 GPa by Chen’s and Tian’s models, which illuminates that the two phases are hard but not superhard.