A tadhyil may have the form of a general maxim that can be cited beyond its immediate context, or it may reinforce a point in wording more closely tied to the specific occasion.
وَهَلْ نُجَازِي إِلَّا الْكَفُورَ
And do We repay with such punishment any but the ungrateful? (Saba 34:17)
Following the preceding account, the statement expresses a more general principle about ingratitude and requital. It moves from a particular event toward a broadly applicable judgment.
“Like a proverb” describes general applicability, not the requirement that the wording already be a popular proverb.