Won't curable STD's eventually mutate and become immune to cures if they keep spreading?

The more a disease spreads from person to person the more it replicates and creates new genes, over time the viruses and bacteria will mutate and become resistant to certain conditions or threats in their environment. If this is the case then won't curable bacterial syds like syphilis or gonorrhea or chlamydia or tricho develop a strain that mutates and doesn't get cured?

Won't curable STD's eventually mutate and become immune to cures if they keep spreading?
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