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The Senate voted Monday along party lines — 60-32 — voted to move along the confirmation if Mrs. Smith. All 58 Democrats — including the one Mr. Brown will replace — and both independents voted to advance the nomination.

Republican senators said Mr. Reid has not violated his pledge to them.

“He’s lived up to that,” said Sen. Lamar Alexander of Tennessee, chairman of the Senate Republican Conference, who said he interpreted Mr. Reid’s vow as applying only to health care votes.

Democrats last week needed all 60 members of their caucus to push through an increase to the country’s debt limit, including the vote of Sen. Paul Kirk, who holds the Massachusetts seat until Mr. Brown is sworn in.

Mr. Alexander said he expects the situation with Mr. Brown to be cleared up any day.