The fact that it gained power was to some degree the fact that it was officially accepted by Constantine, and the fact that Theodosius made it the state religion of the Roman Empire in 380. The Church gained more power as its missionaries converted Germanic invaders to Christianity.
But the Church had a specific power that made it especially powerful, which was that a king or emperor could be excommunicated by the pope, and this meant that all oaths of support made to that monarch and all treaties entered into with that monarch could be broken at will. This meant that rebels and enemies had far more power than they otherwise would. Several kings and emperors tried to cross Church officials, only to discover that they had been gravely weakened in the process. The English Kings Henry II and John, and the Emperor Henry IV of the Holy Roman Empire all suffered from this.
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