Hart, 20, was recalled from Lehigh Valley of the American Hockey League on Monday.
The Flyers (12-15-4) will be the first team in NHL history to start six goalies in the first 35 games of a season and the 14th team to start at least six at any point in a season, the first since the Buffalo Sabres and Edmonton Oilers in 2013-14.
Hart said he learned he was getting called up during Lehigh Valley's Christmas party on Sunday.
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"It's something I've dreamt of as a kid," he said. "I've always wanted to be an NHL hockey player since I was 4 years old, since I started playing hockey. When I got the call last night, I was really excited and kind of shaking with enthusiasm a little bit."
After a bumpy start, Hart, selected by the Flyers in the second round (No. 48) in the 2016 NHL Draft, said he feels more comfortable with the professional game. He's 5-2-0 with a 2.46 goals-against average and .922 save percentage in his past seven games.
He credits the improvement to better practice habits and less thinking about what he needed to do.
"I think just trusting my game, not trying to overthink plays," he said. "Earlier in the year, making the transition from juniors to the pro level, I was over-analyzing things. I had talks with the coaches down there, my goalie coaches back home. I've just got to trust my game, have to play to my strengths and when it's game time it's time to play. You can't think, you just have to play."
Hart had a 1.87 GAA and .922 save percentage in four preseason games with Philadelphia and left an impression on his new teammates.
"He had a strong camp, he looked good in camp, played well for us," defenseman Andrew MacDonald said. "All I know is he's played really well of late, done some good things. He's a big part of this team's future."