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How did the Great Elector end Poland's suzerainty over Ducal Prussia?

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Historical map showing the unification of Germany by Prussia in the late 19th century. The map highlights in green the Kingdom of Prussia as of 1866, and in yellow the additional territories that formed the German Empire by 1871-1918, encompassing most of north-central Europe. Neighboring countries such as France, Belgium, the Netherlands, Denmark, Sweden, Russia, Austria-Hungary, and Switzerland are labeled, with major regions and states within the empire clearly marked. — Prussia The unification of Germany by Prussia, which brought most of north-central Europe into one kingdom.

The Great Elector, Frederick William, ended Poland's suzerainty over Ducal Prussia through a combination of military maneuvering and strategic diplomacy during the First Northern War (1655–1660).12

Initially, Frederick William sought to secure his objective by playing rival powers against one another. After shifting his allegiance between Sweden and Poland, he concluded the Treaty of Wehlau with the Polish king, John Casimir, on September 19, 1657. Under the terms of this agreement, Frederick William pledged to provide Poland with 6,000 troops for use against Sweden. In exchange, the Polish crown formally renounced its suzerainty over Ducal Prussia, recognizing Frederick William and his heirs as the duchy's sovereign rulers.13

These provisions were subsequently confirmed by the Treaty of Oliva in 1660, which officially concluded the conflict. This transition to full sovereignty was a fundamental step in the rise of the Hohenzollern monarchy to the status of a European great power.132