Theorem 2euexv

Description: Double quantification with existential uniqueness. Version of 2euex with x and y distinct, but not requiring ax-13 . (Contributed by NM, 3-Dec-2001) (Revised by Wolf Lammen, 2-Oct-2023)

	Ref	Expression
Assertion	2euexv	⊢ ( ∃! 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	nfmov	⊢ 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 )