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Metamath Proof Explorer

Theorem 2moex

Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2moexv when possible. (Contributed by NM, 3-Dec-2001) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 𝜑
2	1	nfmo	⊢ Ⅎ 𝑦 ∃* 𝑥 ∃ 𝑦 𝜑
3		19.8a	⊢ ( 𝜑 → ∃ 𝑦 𝜑 )
4	3	moimi	⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∃* 𝑥 𝜑 )
5	2 4	alrimi	⊢ ( ∃* 𝑥 ∃ 𝑦 𝜑 → ∀ 𝑦 ∃* 𝑥 𝜑 )