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Let denote the following property of a variety : Every subquasivariety ofis a variety. In this paper, we prove that every idempotent dual discriminator variety has property . Property need not hold for nonidempotent dual discriminator varieties, but does hold for minimal nonidempotent dual discriminator varieties. Combining the results for the idempotent and nonidempotent cases, we obtain that every minimal dual discriminator variety is minimal as a quasivariety