Rai, who was a nominee of the bank's infra lender parent IDFC Ltd on the Board, stepped down "to avoid any potential conflict of interest" which may arise because of his appointment as the chief of BBB, the bank said in a statement.
He has also served as Comptroller and Auditor General of India.
Sinha, a career central banker, oversaw the last round of bank licensing in which IDFC Bank was selected along with Bandhan to launch universal banks among nearly two dozen hopefuls.
IDFC Bank began operations in last October, two months after microlender-turned-universal bank Bandhan was launched.
Sinha joined law firm Amarchand Mangaldas as a senior advisor and has also been associated with private equity major KKR as an industry advisor.
He joined the RBI in 1976, and was in-charge of regulation of commercial banks, Non-Banking Financial Companies, Urban Cooperative Banks and Information Technology, the statement said.
"He has been closely associated with the banking sector reforms in India," it said.
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