See, in the books he plays with all aspects of it.
The Witch King dies in the books, not because Éowyn is a woman, but due to compounding factors.
After the Battle of Fornost, Glorfindel prophesied that the Witch King would not fall by the hand of a man, due to this, the he believed himself to be essentially invincible in combat.
Early in the story, shortly after leaving the Shire, the Hobbits are captured by a spirit called a Barrow-wight.
The spirit traps them inside an ancient burial mound, and prepares to sacrifice them, Frodo manages to call for help, and Tom Bombadil arrives to banish the spirit.
While sorting through the mound’s treasure to break the tomb's curse, Bombadil finds four long, leaf shaped daggers.
Because of the hobbits' small stature, these daggers serve perfectly as short swords, Tom gifts one to each hobbit.
Tom explains that these Barrow-blades were forged by the Dúnedain of Arnor, crafted specifically for their wars against their greatest enemy, the Witch King of Angmar.
During the Battle of the Pelennor Fields, Merry had ambushed the Witch King and stabbed him behind the knee.
He uses a Barrow-blade, the magical dagger breaks the dark enchantments that knit the Witch King's unseen body to his will, effectively breaking his invulnerability.
Distracted, and weakened by Merry's blow, the Witch King is struck in the head by Éowyn.
Because Éowyn is a woman, and Merry is a Hobbit that used the blade of a Wraith, they brought the prophecy to pass.
The, I am no man line is a Jackson insert to simplify the narrative, as he had erased one of the most important characters from the story, and erasing her important monologue from the story too.
The Witch-king: "Hinder me? Thou fool. No living man may hinder me!"
Éowyn: "But no living man am I! You look upon a woman. Éowyn I am, Éomund's daughter. You stand between me and my lord and kin. Begone, if you be not deathless! For living or dark undead, I will smite you, if you touch him."
So yeah, the prophecy is brought to pass because of a Hobbit, a woman, a wraith, and Tom Bombadil.
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