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Later, however, Mr. Biden came down from the chair and laughed with Mr. McConnell on the floor.

GOP senators had blocked Mr. Cordray for two years, arguing they had fears over the newly created Consumer Financial Protection Bureau he was tapped to lead. They figured that by denying the independent agency its head, they could stymie its work.

President Obama then issued a recess appointment for the CFPB nominee at the beginning of last year, along with several picks to the National Labor Relations Board.

A federal appeals court found the NLRB nominations unconstitutional, and the GOP had been insistent that those and the Cordray nomination not be allowed to proceed because it would give a stamp of approval to Mr. Obama’s recess appointment maneuver.

The GOP had been insistent that those nominees and the Cordray nomination not be allowed to proceed because it would give a stamp of approval to Mr. Obama’s recess move. But by this week, Republicans had dropped their objection to Mr. Cordray, focusing instead on the NLRB picks.