[Sorry if the question looks trivial but I have thought about it for days and red a number of books all with no success.]
In almost every thermodynamics book that has something to say about chemical reactions there is a proof for the fact that at constant pressure and temperature the process is spontaneous in the direction in which the Gibbs free energy decreases. In some of the books it is also remarked that this conclusion is for when there is no "additional work" (beside the work of pressure in expanding the system), if there is an additional work then except for $dG=0$ the reversible process will yield into that additional work gaining its maximum value:
\begin{align} \text{The only work is expansive:}\qquad & dG\le 0\\ \text{There are some additional works present:}\qquad & dG\le \delta W_{\text{additional}} \end{align}
In this regard I have a question: When we have some additional works there in the problem (like electric work in an electrochemical cell), so that the second inequality holds and the maximum work is assigned to $dG$ as $dG=\delta W_{Add,Max}$, why still we feel free to use the first inequality and claim $dG$ should be negative?
In the case of electrochemical cells we have: $\delta W_{E,Max}=-n\,F\,E_{cell}\,d\zeta$ wherein $\zeta$ measure the advancement of the reaction. So we would obtain in a reversible (i.e. infinitely slow) reaction $\Delta_rG=\frac{dG_{p,t}}{d\zeta}=-n\,F\,E_{cell}$. But all the books then state that since the reaction is spontaneous in the direction in which $\Delta_rG\le 0$ so we would obtain $E_{cell}\ge 0$, while we know the inequality $\Delta_rG\le0$ is obtained from $dG\le0$ which doesn't hold here.
My question in other wording reads like this. Does the electrochemical cell problem involve an electric work or not? If "yes" then from where do we know the Gibbs free energy should decrease? And if "no" then from where do we obtain the equality $\Delta_rG=-n\,F\,E_{cell}$?
