At the Christian Leadership Center website, I offer some reflections on the priesthood in interpreting John 13:

In John 13 we find parallels to Leviticus 16, the Day of Atonement ritual. Lev 16:23-24 reads, “Then Aaron shall come into the tent of meeting, and shall put off the linen garments which he put on when he went into the holy place, and shall leave them there; and he shall bathe his body in water in a holy place, and put on his garments, and come forth, and offer his burnt offering and the burnt offering of the people, and make atonement for himself and for the people.” Observe the pattern: The high priest undresses, bathes, dresses, and offers sacrifice. In John 13, Jesus undresses (v. 4), washes the disciples’ feet (v. 5-11), dresses (v. 12), and will soon offer himself in sacrifice. Whereas in Leviticus the high priest washes all of himself, in John, Jesus washes the feet of the disciples. Jesus is sharing his high priesthood with the disciples; he must wash them — ordain them as priests — lest they have “no part” in his priesthood.

Indeed, washing seems part of priestly ordination elsewhere in the OT. In the midst of the “consecration” (Lev 8:10) of Aaron and his sons, Moses “washed them with water” (Lev 8:6). We also see Aaron and his sons being washed in Exodus 40:

Then you shall bring Aaron and his sons to the door of the tent of meeting, and shall wash them with water. (v. 12)And he set the laver between the tent of meeting and the altar, and put water in it for washing, with which Moses and Aaron and his sons washed their hands and their feet; when they went into the tent of meeting, and when they approached the altar, they washed; as the LORD commanded Moses. (vv. 30-32)

Furthermore, the mention of a “part” (μέρος) in John 13:8 recalls the Levites having their portion (μερίς) in the LORD (Num 18:20 and Deut 10:9 LXX).

Read the whole thing.