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This rivalry basically stems from the 2010 Stanley Cup Final. Not all Stanley Cup matchups turn into rivalries, and this isn't exactly a heated rivalry, but there is a little bit of tension between the teams because of how past moves have ended up.
For example, the Flyers traded Patrick Sharp to the Blackhawks several years back for Matt Ellison and a third-round pick. They then went on to watch him become one of the top players on the team that beat them in the Stanley Cup Final.
Then, in 2007, the Flyers finished with easily the worst record in the NHL. The Blackhawks, however, won the draft lottery and selected Patrick Kane with the first overall pick. The Flyers watched as Kane scored the Cup-winning overtime goal in Game 6.