Theorem 2eu7

Description: Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 19-Feb-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	2eu7	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 𝜑
2	1	nfeu	⊢ Ⅎ 𝑥 ∃! 𝑦 ∃ 𝑥 𝜑
3	2	euan	⊢ ( ∃! 𝑥 ( ∃! 𝑦 ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ( ∃! 𝑦 ∃ 𝑥 𝜑 ∧ ∃! 𝑥 ∃ 𝑦 𝜑 ) )
4		ancom	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ( ∃ 𝑦 𝜑 ∧ ∃ 𝑥 𝜑 ) )
5	4	eubii	⊢ ( ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ∃! 𝑦 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑥 𝜑 ) )
6		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑
7	6	euan	⊢ ( ∃! 𝑦 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑥 𝜑 ) ↔ ( ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) )
8		ancom	⊢ ( ( ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ↔ ( ∃! 𝑦 ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
9	5 7 8	3bitri	⊢ ( ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ( ∃! 𝑦 ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
10	9	eubii	⊢ ( ∃! 𝑥 ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) ↔ ∃! 𝑥 ( ∃! 𝑦 ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )
11		ancom	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ↔ ( ∃! 𝑦 ∃ 𝑥 𝜑 ∧ ∃! 𝑥 ∃ 𝑦 𝜑 ) )
12	3 10 11	3bitr4ri	⊢ ( ( ∃! 𝑥 ∃ 𝑦 𝜑 ∧ ∃! 𝑦 ∃ 𝑥 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜑 ) )