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Metamath Proof Explorer

Theorem moabs

Description: Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002) Shorten proof and avoid df-eu . (Revised by BJ, 14-Oct-2022)

		Ref	Expression
	Assertion	moabs	⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃* 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( ∃* 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → ∃* 𝑥 𝜑 ) )
2		nexmo	⊢ ( ¬ ∃ 𝑥 𝜑 → ∃* 𝑥 𝜑 )
3		id	⊢ ( ∃* 𝑥 𝜑 → ∃* 𝑥 𝜑 )
4	2 3	ja	⊢ ( ( ∃ 𝑥 𝜑 → ∃* 𝑥 𝜑 ) → ∃* 𝑥 𝜑 )
5	1 4	impbii	⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃* 𝑥 𝜑 ) )