Я желаю тебе здоровья.

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Breakdown of Я желаю тебе здоровья.

• здоровье: the health
• тебе: you
• желать: to wish

Questions & Answers about Я желаю тебе здоровья.

Why is it тебе and not ты?

Ты is the nominative form (used for the subject: “you do something”).
Тебе is the dative form (used for the indirect object: “to you / for you”).

The verb желать (“to wish”) normally follows the pattern:

• желать
  • кого́/чего́ (Genitive: what you wish)
• кому́ (Dative: to whom you wish it)

So in Я желаю тебе здоровья, тебе is dative and means “to you”:
I wish (to you) health.

What case is здоровья, and why is that case used?

Здоровья is in the genitive singular case. The base word is здоровье (“health”).

With желать, the thing being wished is put in the genitive case:

• Желать счастья – to wish (someone) happiness
• Желать удачи – to wish (someone) good luck
• Желать здоровья – to wish (someone) health

So здоровья is genitive because желать requires it:
желать чего? → здоровья.

Can you say Я желаю тебе здоровье with здоровье in the nominative?