Theorem 2eu8

Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put E! on either of the internal conjuncts but not both. We can also commute E! x E! y using 2eu7 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 20-Feb-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	2eu8	⊢ ( ∃! 𝑥 ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		2eu2	⊢ ( ∃! 𝑥 ∃ 𝑦 𝜑 → ( ∃! 𝑦 ∃! 𝑥 𝜑 ↔ ∃! 𝑦 ∃ 𝑥 𝜑 ) )
2	1	pm5.32i	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃! 𝑥 𝜑 ) ↔ ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) )
3		nfeu1	⊢ Ⅎ 𝑥 ∃! 𝑥 𝜑
4	3	nfeu	⊢ Ⅎ 𝑥 ∃! 𝑦 ∃! 𝑥 𝜑
5	4	euan	⊢ ( ∃! 𝑥 ( ∃! 𝑦 ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ( ∃! 𝑦 ∃! 𝑥 𝜑 ∧ ∃! 𝑥 ∃ 𝑦 𝜑 ) )
6		ancom	⊢ ( ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ( ∃ 𝑦 𝜑 ∧ ∃! 𝑥 𝜑 ) )
7	6	eubii	⊢ ( ∃! 𝑦 ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ∃! 𝑦 ( ∃ 𝑦 𝜑 ∧ ∃! 𝑥 𝜑 ) )
8		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑
9	8	euan	⊢ ( ∃! 𝑦 ( ∃ 𝑦 𝜑 ∧ ∃! 𝑥 𝜑 ) ↔ ( ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃! 𝑥 𝜑 ) )
10		ancom	⊢ ( ( ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃! 𝑥 𝜑 ) ↔ ( ∃! 𝑦 ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
11	7 9 10	3bitri	⊢ ( ∃! 𝑦 ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ( ∃! 𝑦 ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
12	11	eubii	⊢ ( ∃! 𝑥 ∃! 𝑦 ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ∃! 𝑥 ( ∃! 𝑦 ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
13		ancom	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃! 𝑥 𝜑 ) ↔ ( ∃! 𝑦 ∃! 𝑥 𝜑 ∧ ∃! 𝑥 ∃ 𝑦 𝜑 ) )
14	5 12 13	3bitr4ri	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃! 𝑥 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
15		2eu7	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
16	2 14 15	3bitr3ri	⊢ ( ∃! 𝑥 ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( ∃! 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )