Case Play 2013-09: Whose Base is it Anyway?

Lightning Case Play: Whose Base is it Anyway? With one out and runners on first and third (R1 Brett Lawrie,

Iassogna declares a double play at third base.

R3 Jose Reyes), batter Adam Lind hits a come-backer to pitcher Hiroki Kuroda, who throws to catcher Chris Stewart, who runs R3 Reyes back towards third base before throwing to third baseman Mark Reynolds. As R3 Reyes dives back towards third, R1 Lawrie comes from first to step on the base. F5 Reynolds tags R1 Lawrie followed by R3 Reyes, resulting in a double play call from 3B Umpire Dan Iassogna.

Question: What happened? Who is entitled to occupy third base at all stages of this play? What rules apply and how is this a double play?

*A lightning case play is a "simple" scenario worth one point. This case play expires Saturday.

Video: Yankees score inning-ending double play with a timed tag at third in Toronto (MLB Must C)