I'm sure Atkins doesn't say that; are you reading him correctly?
G = n1μ1 + n2μ2
At equilibrium G is a minimum (not zero), (dG/dn1)n1+n2 = 0.
μ1dn1 + μ2dn2 + n1dμ1 + n2dμ2 = 0
From the Gibbs-Duhem equation (answer B to the previous question on your picture, whatever it was) Σnidμi = 0 at constant T,P; and dn2 = -dn1.
Hence μ1 - μ2 = 0
Chemical potential is an intensive property which is equal in substances in chemical equilibrium - there is no tendency for reaction to go in either direction because ΔG for a small deviation is zero. In this it is analogous to other "potential" quantities, e.g. temperature (which can be regarded as thermal potential) is the same in two bodies at thermal equilibrium, and heat won't flow between them; charge won't flow if there is no electric potential gradient, etc.