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He's laying himself good odds (>4:1 when he's ~36% to beat 2 pair) with the push and may possibly scare villain off his draw. Only thing that nails hero is a set. As played, hero is pretty much committed to pushing the turn, given the slim amount behind.
As far as playing the flop aggressively, I think it's a reasonable move. If you put villain on a draw (the 7c8c makes straight and flush draws a real possibility) than hero's overpair aggression is +EV. Plus, minraising in position is a very common semibluff-with-a-draw move (to get a free river, I guess). After hero pot commits himself, and villain merely smooth calls, I think it's safe to say that two-pair is out (why wouldn't villain just shove?).
Villain was rope-a-doping with his minraise (and had a set) or was transparently semi-bluffing (with an oesd or flush draw). Is he more likely to have 22/77/88 or 56/9T/TJ/Acxc? If you thought he was drawing well done.
Since you pot-committed yourself on the flop, the turn card was irrelevant, but that was a terrible card. (56,TJ).
All in all I think you played it well. Given an overpair on this flop, you either have to push hard or fold. Calling is off the table. He's got a draw or a set. If he had a draw, you denied him proper odds (+ev for you). If he had a set, whatever, you have to pay it off. Paying off a set with an overpair on such a dry, raggy board as this one is pretty much inevitable.
Edit: ditto for larger overpairs... if he slowplayed JJ+ then this flop was a godsend.
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