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Hold on to the word of life
"Hold on to" here means "firmly believe." Alternate translation: "Continue to firmly believe the word of life" (See: rc://en/ta/man/jit/figs-metaphor)
the word of life
"the message that brings life" or "the message that shows how to live the way God wants you to"
on the day of Christ
This refers to when Jesus comes back to set up his kingdom and rule over the earth. Alternate translation: "when Christ returns"
I did not run in vain or labor in vain
The phrases "run in vain" and "labor in vain" here mean the same thing. Paul uses them together to emphasize how hard he has worked to help people believe in Christ. Alternate translation: "I did not work so hard for nothing" (See: rc://en/ta/man/jit/figs-parallelism)
run
The scriptures often use the image of walking to represent conducting one's life. Running is living life intensely. (See: rc://en/ta/man/jit/figs-metaphor)