First of all, I do not trust anyone at the poker table. If someone declares a hand, I ask to see it.
Also, when the player said that he had a six, he actually had two of them (on the board).
Some rooms have a rule that kills the hand of a player who misdeclares a hand at showdown, apparently with the intent to deceive an opponent.
“Cards speak (cards read for themselves). The dealer assists in reading hands, but players are responsible for holding onto their cards until the winner is declared. Although verbal declarations as to the contents of a hand are not binding, deliberately miscalling a hand with the intent of causing another player to discard a winning hand is unethical and may result in forfeiture of the pot.”
And, lastly, the player who folded did a stupid thing by not just tabling his hand, or folding after seeing a better hand.