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That sent New Zealand into their second straight World Cup final with a surprise 18-run win over India at Old Trafford.
After New Zealand set them a modest target of 240 to win, India were reduced to 24-4 with Rohit Sharma, skipper Virat Kohli and KL Rahul dismissed for a single run each as the Black Caps' opening attack delivered in style.
Matt Henry did the damage with three of the early wickets while left-armer Trent Boult also bowled superbly, keeping the pressure and picking up the vital wicket of India skipper Kohli.
Boult later removing dangerman Ravindra Jadeja, whose 77 off 59 balls had threatened to thwart the Kiwis.
"We spoke about the conditions during the interval, we wanted to put the ball in good areas and move the ball around and put some pressure on India, they are a world-class side," said Williamson.
"With the new ball, our bowlers tried to move the ball off the seam or in the air. We knew we need to put pressure on that world class batting line-up of India
"It was a great start from the bowlers, we knew it would get tougher for us as the innings went on. We showed plenty of heart and the fielders and bowlers were outstanding."
Two of the key moments in the game were bits of brilliance in the field - Jimmy Neesham's astonishing left-handed catch to dismiss Dinesh Karthik for the fourth wicket and a direct hit from Martin Guptill running out MS Dhoni to fatally wound India's late run chase.
Henry, who finished with figures of 3-37, said the performance in the field and with the ball showed the team's spirit.
“I thought we managed to scrap it out. We had the belief - we knew we had to bowl well," he said.
“Obviously, they are world-class batsmen - Hardik, Dhoni and Jadeja - we knew to win the game we had to get them out," he said.
The Kiwis had lost their last three games in the group stage, which was topped by India, who themselves suffered only one defeat.
But Williamson said he was not surprised to see his team come through to set up a final on Sunday against England or Australia.
"You come to this tournament where anybody can beat anybody. We saw a number of results that perhaps where sides ranked lower beat sides ranked higher, but I think everybody knew that anybody can beat anybody," he said.
"We came through in the fourth position, which is fine. And then we put out a much-improved performance in the semi-final and it is important for us to try and make small improvements moving into our next match," Williamson added.