Let's not forget, a third runway at Heathrow was approved previously.
BAA at that point owned both Heathrow and Gatwick and chose to expand LHR. BAA was then sold to a Spanish company and suddenly, it wasn't seen as a good idea to allow BAA to own both LHR & LGW and they were made to sell LGW.
The new owners of Gatwick then realised that they could expand their asset with help from the taxpayer and came up with a plan which they said, could rival LHR. We then had this 'Third runway needed anywhere near London' debate which is meaningless as expansion of the two airports is based on entirely different principles.
What is true is that and expanded Heathrow would bring a hub and relieve the pressure on Gatwick. An expanded Gatwick would do nothing to help LHR's case.