Right, but it still takes a vote by both the house and the senate.
No. The Senate has already passed it’s version of the bill. If the House passes the same exact version, i.e., the Slaughter Rule of deeming that the House has passed it, then there is no need to go back to the Senate for another vote. The only thing that goes back to the senate for a vote is the reconciliation bill that contains changes to the Senate bill. Obama can immediately sign the Senate bill after the House vote.