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Metamath Proof Explorer

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 GG, 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 imbitrrdi ⊢ ( ∀ 𝑥 ∃* 𝑦 𝜑 → ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑦 ∃ 𝑥 𝜑 ) )