Rules question on showing cards

I played in a live MTT this weekend. I had AA and led the betting all the way to the river at which point the river card gave a straight on the board, and I checked. I was called by one player, except on the river when the straight card came up and I checked he bet and I called. He showed 88 and was playing the board. I declared that I was playing the board. He said that he wanted to see my cards. I said fine, but I would ask to see his the rest of time if that's how he wants to play. Someone at the table said that I cannot collect on the pot unless I show my cards. I disagree since I declared I was playing the board. What is correct by the rules in a situation like this?