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Metamath Proof Explorer

Theorem 2euex

Description: Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2euexv when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

Ref Expression
Assertion 2euex ( ∃! 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑦 ∃! 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 df-eu ⊢ ( ∃! 𝑥 ∃ 𝑦 𝜑 ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃* 𝑥 ∃ 𝑦 𝜑 ) )
2 excom ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 𝜑 )
3 nfe1 ⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑
4 3 nfmo ⊢ Ⅎ 𝑦 ∃* 𝑥 ∃ 𝑦 𝜑
5 19.8a ⊢ ( 𝜑 → ∃ 𝑦 𝜑 )
6 5 moimi ⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑥 𝜑 )
7 moeu ⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) )
8 6 7 sylib ⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) )
9 4 8 eximd ⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ( ∃ 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑦 ∃! 𝑥 𝜑 ) )
10 2 9 biimtrid ⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑦 ∃! 𝑥 𝜑 ) )
11 10 impcom ⊢ ( ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃* 𝑥 ∃ 𝑦 𝜑 ) → ∃ 𝑦 ∃! 𝑥 𝜑 )
12 1 11 sylbi ⊢ ( ∃! 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑦 ∃! 𝑥 𝜑 )