"Governing law" (or "proper law") is the law that decides the substantive rights and obligations under the contract — who owes what to whom, and why. "Procedural law" (usually the law of the seat) governs how the arbitration itself is conducted — appointment of arbitrators, interim relief, and challenges to the award.
In most domestic Indian contracts these coincide, and few practitioners think twice. In cross-border contracts they routinely diverge: a contract governed by English law can still be arbitrated with its seat in Chennai, applying Indian procedural law to the arbitration while English law decides the merits.
A clause that names only one "governing law" for everything, without addressing the seat separately, leaves this split unresolved. State both explicitly, and confirm they work together rather than assuming a single line covers everything.