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If a Fighter 3 (Eldritch Knight) levels up as a Warlock for the rest of the way to 20, making him an 18th level caster according to the multiclass spell slot table provided, would he in turn be able to learn 9th level spell? Meeting only the requirements listed for Warlock upon leveling, being 1 spell on it's list for which it has spell slots available for.

(Note: I would put up quotes and annotations providing source material but I'm running on limited time and my question sounds convoluted and simple to solve to the point where it's just me being dumb.)

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I understand your question, but I'm afraid you've made a mistake in your calculations. An Eldrtich Knight 3/Warlock 17 does not have a caster level of 18 for the purposes of multiclass spellcasting. It has two separate kinds of casting: level 3 Eldritch Knight casting and level 17 Warlock casting. As it says on PHB 164:

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature.

Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.

Note that in the Spell Slots list, Warlock is not listed. You don't add together your EK and Warlock levels, because one gives you Pact Magic and the other gives you Spellcasting, which don't stack.

If you're an Eldritch Knight 3/Warlock 17, then you can cast EK spells using Warlock slots and Warlock spells using EK slots, but neither kinds of casting are improved by having the other class.

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