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Metamath Proof Explorer

Theorem moexex

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 . Use the version moexexvw when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 28-Dec-2018) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	moexex.1	⊢ Ⅎ 𝑦 𝜑
	Assertion	moexex	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∀ 𝑥 ∃* 𝑦 𝜓 ) → ∃* 𝑦 ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		moexex.1	⊢ Ⅎ 𝑦 𝜑
2	1	nfmo	⊢ Ⅎ 𝑦 ∃* 𝑥 𝜑
3		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ( 𝜑 ∧ 𝜓 )
4	3	nfmo	⊢ Ⅎ 𝑥 ∃* 𝑦 ∃ 𝑥 ( 𝜑 ∧ 𝜓 )
5	1 2 4	moexexlem	⊢ ( ( ∃* 𝑥 𝜑 ∧ ∀ 𝑥 ∃* 𝑦 𝜓 ) → ∃* 𝑦 ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )