It depends on what temperature is. A simple statistical argument could go like this.
- the multiplicity of two systems is the product of their multiplicities
- in thermodynamic equilibrium the multiplicity is at its maximum
- the fractional change of multiplicity with internal energies ($\beta = \frac{1}{\Omega} \frac{d\Omega}{dE}$) is equal for both systems.
- identify $\beta = 1/kT$
If this is the case for systems A and B and for B and C, the fractional change of multiplicity with energy ($\beta$) should also be the same for A and C.