Your vet is wrong. Is he an older man? Waiting until after the first heat is very outdated information.
Spaying before her first heat will eliminate her chance of ever getting mammary cancer. If you wait until after her first heat, the risk rises to 8%. If you wait until after she has her second heat, her risk is as great as an unspayed dog, about 25%.
R - Reproduction - When to Spay The Canine Spay