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I am sorry to hear about the severe allergies that your daughter has.
At 6 months, egg allergy is unlikely to produce these symptoms, as the child is not consuming eggs.
Cow milk allergy is certainly a possibility though, and a trial of soy milk may be useful in this situation.
I would recommend getting skin / blood tests for other allergens, and then trying to eliminate all environmental allergens as well.
It may be helpful to consult an allergy specialist for this.
Finally, using higher potency steroids, and a short course of oral steroids maybe needed to control the symptoms, along with long-term use of oral antiallergic medicine like cetirizine so that her symptoms are controlled,