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	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to {\exists }^{*}x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢{\exists }^{*}x\phi \to \left(\exists x\phi \to {\exists }^{*}x\phi \right)$$
2		nexmo	$$⊢\neg \exists x\phi \to {\exists }^{*}x\phi$$
3		id	$$⊢{\exists }^{*}x\phi \to {\exists }^{*}x\phi$$
4	2 3	ja	$$⊢\left(\exists x\phi \to {\exists }^{*}x\phi \right)\to {\exists }^{*}x\phi$$
5	1 4	impbii	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to {\exists }^{*}x\phi \right)$$