Your AC system has filters specially designed to remove dust and other impurities. These filters remove almost all possibilities for allergies to exist. Since the air conditioning also removes excess humidity from the air, it prevents the build-up of mould and bacterial growth. This helps improve your home’s air quality.
People that report being allergic to AC are typically allergic to what's in the air being circulated. This is why proper filters or air cleaners are essential for AC and furnace systems. Your home’s comfort extends beyond just the air temperature/humidity, to the cleanliness of it too.