Bronsted Neutralization Reaction

[chloewinnett1L]

The solutions manual says that the proper redox reaction for the formation of water from hydronium and hydroxide ions involves the two following reactions:
(cathode) and (anode).
Using these two reactions gives you an overall standard cell potential of +0.83V.

However, I used the following two reactions to obtain water:
(cathode) and (anode)
and ended up with the exact same standard cell potential, +0.83V.

My question is, are both of these routes valid? As far as I can tell, the only difference between the two is that the first path uses oxygen gas as an intermediate and the second path uses hydrogen gas as an intermediate.

[Tiffany_Cacy_3D]

I did the same exact thing and got the same answer. I think as long as everything cancels out correctly it should be the same. On the test there will only be a certain number of reactions to choose from as well so I think it will be more clear which reactions we are supposed to use.