Moses 5:4 says: “Adam and Eve, his wife, called upon the name of the Lord.”
Moses 6:4, 7 says: “And then began these men to call upon the name of the Lord … Now this same Priesthood, which was in the beginning, shall be in the end of the world also.”
I think there is a lot going on here in these three simple verses. What Adam and Eve do (call on the name of the Lord together), is later referred to as “Priesthood” when a group of men do it.
What do we make of that?
Here are some ideas:
1) Here, way back in the book of Moses, in taking about the first man and woman, I think I see the same situation I’ve wondered about recently: women can do Priesthood-like things without an ordination to the Priesthood (i.e., temple endowment, missions, leadership callings)
2) What Adam and Eve do comes first, and Priesthood comes second.
3) The Priesthood seems to be a matter of generations, while Eve-Adam version isn’t?
4) From #3, could we say that the Priesthood is vertical (multi-generational) while the Eve-Adam version is horizontal (immediate, temporal, lateral)?
5) D&C 128 links Priesthood unmistakably to uniting generations.
6) Why today do we have a presidency consisting of 3 men of any generation or family, whereas with Adam, he needed 3 generations? What could we learn about the priesthood from that?
7) Did that change with Abraham? Are we all considered Abraham’s seed, and so any three men will do?
8) Did Abraham’s covenant change anything for women? Do we still operation with the Eve-Adam version?
9) What about women not married? Eve of course was, so I don’t see what we can learn directly there. However, it’s incredibly important to remember that women, married or not, can receive temple endowments, serve missions, and serve in leadership without Priesthood ordination. (Note that when we had a “Priesthood ban” for black member, they could not receive endowments or sealings. The situation for women is actually quite different.)
10) What other things should I note from these 3 verses?