Question about the 0th law of thermodynamics

[QuasarBoy543298]

assume I have 3 systems a,b and c.
the equilibrium between a and c can be described by the equality P_cV_c = f(A1,..., An)
and the same for b and c - P_cV_c = g(B1,...,Bk).
why does the equality g(B1,..., Bk) = f(A1,..., An) describes an equilibrium between A and B?
the 0^th law states that if A and B are in equilibrium with c **for the same value of the thermodynamics coordinates of C**
then they are in equilibrium with each other, but I can find different pairs of P_c and V_c that will result in the same value of P_cV_c.

 

[Dale]

Isn’t PV proportional to T? So this just describes three systems with the same temperature.

 

[QuasarBoy543298]

I'm not talking about ideal gas but on general systems.
you can take another combination as well, such as PN, and the question stays the same.

 

[Lord Jestocost]

A persuasive representation which – to my mind – addresses your question can be found in “The Principles of Chemical Equilibrium“ (see pages 9-13) by Kenneth Denbigh.
https://archive.org/details/ThePrinciplesOfChemicalEquilibrium/page/n30