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Initially, two Democrats — Sens. Ben Nelson of Nebraska and Mark Pryor of Arkansas — joined Republicans in voting against the rules change, and a third, Sen. Claire McCaskill of Missouri, seemed poised to follow suit.

But a visibly incensed Mr. Reid found Mr. Pryor on the GOP side of the chamber and, gesturing emphatically at Mr. Pryor, argued his case. Meanwhile, other Democrats pleaded with Mrs. McCaskill.

In the end, both switched their votes to back their party, leaving Mr. Nelson and all 47 Republicans in opposition.