How can I translate this quantified logical expression into english
The negation is that every woman has an airline that she has never flown with. This is equivalent to saying that, for every woman there's an airline with the following property:


for every flight in the whole world, either the woman was not on that flight (regardless of what airline operated it) or the flight was not operated by that airline (regardless of whether the woman was on it or not).


That is, there was no flight by the airline in question that the woman was on.

This is not quite the same as what you have.