The account of foot-washing only occurs in John's gospel, so we need to establish our justification for making it part of the Passover ceremony or not.
Lets take a close look at the scripture in question and see what exactly it says on the subject.
John 13:4-17 "He (Christ) riseth from supper, and laid aside his garments; and took a towel, and girded himself. After that he poureth water into a basin, and began to wash the disciples' feet, and to wipe them with the towel wherewith he was girded. Then cometh he to Simon Peter: and Peter saith unto him, Lord, dost thou wash my feet? Jesus answered and said unto him, What I do thou knowest not now; but thou shalt know hereafter (he will explain after he does it). Peter saith unto him, Thou shalt never wash my feet . Jesus answered him, If I wash thee not, thou hast no part with me. Simon Peter saith unto him, Lord, not my feet only, but also my hands and my head. Jesus saith to him, He that is washed needeth not save to wash his feet, but is clean every whit: and ye are clean, but not all. For he knew who should betray him; therefore said he, Ye are not all clean. So after he had washed their feet, and had taken his garments, and was set down again, he said unto them, Know ye what I have done to you? Ye call me Master and Lord: and ye say well; for so I am. If I then, your Lord and Master, have washed your feet; ye also ought to wash one another's feet. For I have given you an example, that ye should do as I have done to you. Verily, verily, I say unto you, The servant is not greater than his lord; neither he that is sent greater than he that sent him. If ye know these things, happy are ye if ye do them."
After washing their feet Christ explains that yes he is their Master, makeing the disciples his servants.
"Ye call me Master and Lord: and ye say well; for so I am."
At that time in history, it was the servants responsibility to wash their Masters feet when they arrived at home.
Christ goes on to explain that he washed their feet to take on the roll of servant this was to show the state of mind they should have one towards another.
"If I then, your Lord and Master, have washed your feet;"
"For I have given you an example, that ye should do as I have done to you. Verily, verily, I say unto you, The servant is not greater than his lord;"
Could Christ have chosen to bring this up because he knew that his time with them was up, and that there might be dissention between them as to who should lead in Christ's place once gone?
Luke 22:24-30 "And there was also an argument among them, even this: which of them should be considered the greatest. And He said to them, “The kings of the nations rule over them, and those who exercise authority over them are called benefactors. But it shall not be this way among you; rather, let the one who is greatest among you be as the younger, and the one who is leading as the one who is serving. For who is greater, the one who is sitting at the table, or the one who is serving? Is not the one who sits at the table? But I am among you as one who is serving. ...”
The point he was making was that they are all equal and should serve one another.
"ye also ought to wash one another's feet."
"If ye know these things, happy are ye if ye do them."
Know what? Do what? Serve one another! Again he reminds them of this in John15:20
"Remember the word that I said unto you, The servant is not greater than his lord."
Christ’s illustration of washing feet is focusing on:
1) Obedience, By not letting Christ wash his feet Peter was disobeying.
2) Spiritual cleaning, if you have bathed (accepted Christ) you are clean except your feet, there is always sin in a humans life.
3) Humility, One is not greater than the other.
If this was to be instituted into the ceremony would not one of the others present have said so in later writings? And why is there no mention of it in any historical writings of non Christians?
Clearly Christ was not instituting another ceremony, but providing an example of Christian attitude.
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