This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 2eu1

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2eu1v when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 23-Apr-2023) (New usage is discouraged.)

Ref Expression
Assertion 2eu1 ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 ↔ ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 2eu2ex ⊢ ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )
2 moeu ⊢ ( ∃* 𝑦 𝜑 ↔ ( ∃ 𝑦 𝜑 → ∃! 𝑦 𝜑 ) )
3 2 albii ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 ↔ ∀ 𝑥 ( ∃ 𝑦 𝜑 → ∃! 𝑦 𝜑 ) )
4 euim ⊢ ( ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∀ 𝑥 ( ∃ 𝑦 𝜑 → ∃! 𝑦 𝜑 ) ) → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃! 𝑥 ∃ 𝑦 𝜑 ) )
5 3 4 sylan2b ⊢ ( ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∀ 𝑥 ∃* 𝑦 𝜑 ) → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃! 𝑥 ∃ 𝑦 𝜑 ) )
6 5 ex ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃! 𝑥 ∃ 𝑦 𝜑 ) ) )
7 1 6 syl ⊢ ( ∃! 𝑥 ∃! 𝑦 𝜑 → ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃! 𝑥 ∃ 𝑦 𝜑 ) ) )
8 7 pm2.43b ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃! 𝑥 ∃ 𝑦 𝜑 ) )
9 2euswap ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃ 𝑦 𝜑 → ∃! 𝑦 ∃ 𝑥 𝜑 ) )
10 8 9 syld ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃! 𝑦 ∃ 𝑥 𝜑 ) )
11 8 10 jcad ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 → ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ) )
12 2exeu ⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) → ∃! 𝑥 ∃! 𝑦 𝜑 )
13 11 12 impbid1 ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃! 𝑥 ∃! 𝑦 𝜑 ↔ ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ) )