The Proteas wicketkeeper struck 37 off 27 balls, including four fours and a six, as Leeds posted 187-5 at Headingley. Rickelton shared a 111-run opening partnership with Mitchell Marsh, who provided the attacking impetus with 69 off 34 balls.
Adil Rashid dismissed both openers in a spell that saw him get 3-21, but Dan Lawrence’s unbeaten 51 off 25 balls carried Leeds to a strong total.
Southern Brave remained in the contest throughout their chase before finishing on 182-5. Stubbs led the late charge with an unbeaten 43 off only 20 balls, striking three fours and three sixes.
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With Brave needing 21 runs from the final five deliveries, Stubbs hammered a four and a six to keep the contest alive. Matthew Potts held his nerve, however, as the hosts secured their first victory of the campaign.
David Miller made nine off seven balls before falling to Abrar Ahmed. He had earlier caught Zak Crawley for a golden duck.
Meanwhile, Marco Jansen’s Welsh Fire beat MI London by 15 runs in Cardiff. Jansen struck four off two balls as Fire posted 167-5, with Rachin Ravindra blasting 98 off 44 balls.
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