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 ) ) )