Remember what I told you (so also New American Bible) is literally “remember the word that I said to you” (Revised Standard Version). The reference is to the words of Jesus that follow (No slave is greater than his master), and so New English Bible translates “Remember what I said,” while Jerusalem Bible renders “Remember the words I said to you.” No slave is greater than his master refers back to 13.16. The emphasis in Chapter 13 is on imitating the humility of Jesus, while here it relates to the necessity of sharing his fate.
Remember does not necessarily imply here that the disciples had forgotten what Jesus had said. An appropriate equivalent may be “Think about what I told you.”
In this context greatness may be interpreted as a matter of importance, for example, “A slave is not more important than his master” or “No slave surpasses his master in importance.” In some languages the relation may be inverted by saying “A master is always more important than his slave.”
If they persecuted me is a condition true to fact, as was the “if” clause in verse 18. The meaning is that they did persecute Jesus, and so they will also persecute his disciples. Though the clause If they persecuted me is formally a condition, it states a fact, and in combination with the following clause must be somewhat altered in some languages, for example, “They persecuted me, and they will persecute you too” or “they caused me to suffer, and accordingly, they will cause you to suffer also.”
If they obeyed my teaching is literally “if they kept my word.” But the use of the verb “to keep” in this context has the meaning of “to obey”; and “the word” of Jesus is a summary term for his teaching.
Throughout verses 18-20 the description of the world’s attitude has been completely negative, but here Jesus suddenly seems to change; there are some who obey his teaching, and so there will also be some who will obey the teaching of his followers. That is, Jesus is indicating that the attitude of the world toward his disciples will not be completely negative, just as it was not completely negative toward him. Some did respond to his teaching (for example, the disciples themselves), and so some will also respond to the teaching of his followers. Some commentators assume a negative implication behind Jesus’ words here, for example, New English Bible “they will follow your teaching as little as they have followed mine.” This interpretation is apparently the basis for Die Bibel im heutigen Deutsch: “as little as they have believed my word, so shall they believe yours.” However, even though one might like to read this meaning into the present passage, especially in view of the negative implications in verse 21, it is difficult to arrive at this meaning on the basis of the Greek sentence structure, and so it is probably better not to translate with negative implications. Jesus’ words should be taken to mean that if there are some who persecuted him, there were also some who listened to him; and accordingly, if there are some who persecute the disciples, there will also be some who will respond to their message. One may, therefore, translate the second part of verse 20 “Some persecuted me, and they will persecute you also, but others obeyed what I said, and they will obey also what you say.”
Quoted with permission from Newman, Barclay M. and Nida, Eugene A. A Handbook on the Gospel of John. (UBS Handbook Series). New York: UBS, 1980. For this and other handbooks for translators see here .