(This goes as well for champagne, red wine and even rosé!) Drinking wine is known to erode the enamel and this is because of the acids present in wines that are able to work themselves deep into the tooth.
With white wines, stains from other foods and beverages are highlighted and come on stronger giving the urge to want to brush the teeth but comes with more harm than good!
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If you brush the teeth immediately after drinking white wine (or red or any other acidic drink for that matter) the enamel is eroded instead let the teeth be for about 45 minutes to an hour before trying to clean it to allow it 're-mineralize'.
Instead, opt for swishing the mouth with water in between drinks, when you swish the mouth with water in between drinking, the acid present in wines won't settle on the teeth and if you still want to go ahead to brush afterwards, you can do this as well but to be safe, do this after 30 minutes!