Courts found that the North Carolina voter ID laws were designed to be discriminatory[1]. Same in Texas[2]. So it's definitely true in at least two cases that voter ID laws were provably designed for voter suppression.
Something being provably true is a higher bar than it being explicitly stated (because an explicit statement isn't enough to prove something to be true). I agree that voter suppression was the primary intent, but the legislation wasn't called "The Illegal Voter Suppression Act 2013".
[1]: https://www.theatlantic.com/politics/archive/2017/05/north-c... [2]: https://www.theguardian.com/us-news/2016/jul/20/texas-voter-...