Bravo led his nearest rival in the jockey standing, 61 victories to 59, after tonight's program.
Bailey's lead over Prado in the jockey standing remains at seven.
Bailey, who has won more than $16 million in purses this year to top the national jockey standing, has long been considered a money rider.
He is 12th in the jockey standings at the Belmont Park meet.
Smith, who has been riding back toward the top of the jockey standings after a slow start, was just as awed by the performance.
In the jockey standing, Jerry Bailey and John Velazquez are tied with 11 victories.
Santos found the victory all the sweeter since he was beating Cordero and the triumph gave him a long lead in the jockey standings.
He stayed at the top of the jockey standings, but said he worried about pressing his luck.
He will be teamed again with Jerry Bailey, who is leading the jockey standings at Belmont.
Unable to crack the top 10 in the jockey standings at the Santa Anita meet this winter, Desormeaux had enough.