nexmo

Description: Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022) Avoid ax-11 . (Revised by Wolf Lammen, 16-Oct-2022)

		Ref	Expression
	Assertion	nexmo	$⊢ \neg \exists x φ \to \exists^{*} x φ$

Step	Hyp	Ref	Expression
1		pm2.21	$⊢ \neg φ \to (φ \to x = y)$
2	1	alimi	$⊢ \forall x \neg φ \to \forall x (φ \to x = y)$
3	2	alrimiv	$⊢ \forall x \neg φ \to \forall y \forall x (φ \to x = y)$
4	3	19.2d	$⊢ \forall x \neg φ \to \exists y \forall x (φ \to x = y)$
5		alnex	$⊢ \forall x \neg φ \leftrightarrow \neg \exists x φ$
6	5	bicomi	$⊢ \neg \exists x φ \leftrightarrow \forall x \neg φ$
7		df-mo	$⊢ \exists^{*} x φ \leftrightarrow \exists y \forall x (φ \to x = y)$
8	4 6 7	3imtr4i	$⊢ \neg \exists x φ \to \exists^{*} x φ$

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