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How did Frederick William of Brandenburg gain sovereignty over Prussia in 1657?

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Map showing Poland's territory in 1634 under the reign of Wladyslaw IV Vasa, shaded in pink and spanning from the Baltic Sea in the north to near Kyiv in the southeast. Major present-day cities such as Warsaw, Krakow, Gdansk, Vilnius, Minsk, Lwow (Lviv), Pinsk, and Kyiv are marked within the territory, with surrounding cities like Moscow, Riga, Vienna, and Budapest shown for reference. — Polish-Lithuanian Commonwealth Poland's territory in 1634, during the reign of Władysław IV Vasa.

Frederick William, the Elector of Brandenburg and Duke of Prussia, secured sovereignty over the Duchy of Prussia in 1657 by adroitly maneuvering between the competing powers of Sweden and Poland during the First Northern War.12

Initially, Frederick William sought to protect his interests by maintaining armed neutrality, but he was eventually forced to accept Swedish suzerainty after Charles X Gustav of Sweden overran the region. However, as Sweden’s military position weakened, Frederick William entered into negotiations with the Polish king, John II Casimir Vasa.13

Through the Treaty of Wehlau (and the subsequent Treaty of Bromberg) signed in 1657, Frederick William agreed to provide Poland with 6,000 troops to fight against Sweden. In exchange, King John Casimir renounced Polish suzerainty over ducal Prussia and formally recognized Frederick William and his heirs as the territory's sovereign rulers. This status was later confirmed by the Treaty of Oliwa in 1660, which concluded the war.321