Theorem 2moswapv

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by Gino Giotto, 22-Aug-2023) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023)

	Ref	Expression
Assertion	2moswapv	⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑦 ∃ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑
2	1	nfmov	⊢ Ⅎ 𝑦 ∃* 𝑥 ∃ 𝑦 𝜑
3		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ 𝜑 )
4	3	nfmov	⊢ Ⅎ 𝑥 ∃* 𝑦 ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ 𝜑 )
5	1 2 4	moexexlem	⊢ ( ( ∃* 𝑥 ∃ 𝑦 𝜑 ∧ ∀ 𝑥 ∃* 𝑦 𝜑 ) → ∃* 𝑦 ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ 𝜑 ) )
6	5	expcom	⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑦 ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ 𝜑 ) ) )
7		19.8a	⊢ ( 𝜑 → ∃ 𝑦 𝜑 )
8	7	pm4.71ri	⊢ ( 𝜑 ↔ ( ∃ 𝑦 𝜑 ∧ 𝜑 ) )
9	8	exbii	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ 𝜑 ) )
10	9	mobii	⊢ ( ∃* 𝑦 ∃ 𝑥 𝜑 ↔ ∃* 𝑦 ∃ 𝑥 ( ∃ 𝑦 𝜑 ∧ 𝜑 ) )
11	6 10	syl6ibr	⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑦 ∃ 𝑥 𝜑 ) )