The washing of the disciples’ feet is a symbol with four lessons, and all of them are of tremendous importance. First, it is a symbol of love.
Why does John omit any references to the Lord’s Supper? Simply because John’s Gospel is the last to be written. The Gospel of John does not trouble to tell us about the Lord’s Supper because we are already familiar with it from the other three gospels. Some people have tried to say that the Gospel of John has two suppers. Others have said the Gospel of John contradicts the other Gospels because he has the supper a day earlier than the others. They read in connection with Judas again in John 13:29, ‘Some of them [the disciples] thought, because Judas had the bag, that Jesus had said unto him, Buy those things that we have need of against the feast’, and they leap on this verse, and they say, You see, the Passover feast has not happened yet. John has another meal, another supper. What they do not grasp is that there in verse 29, ‘the feast’ refers to the whole seven-day feast. So the Passover is eaten, verse one, and during the course of the Passover, Judas is sent out, the disciples think mistakenly that Christ has told Judas to go and buy things for the feast. But there are six more days, so it is not surprising they might suppose that. He had not in fact asked any such thing. But there were six more days of this feast of Passover, the crucial day being the first one. So there is no contradiction. Whenever the feast is referred to, it always refers to the whole feast, not just the supper, the Passover night.