Theorem 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	$$⊢\neg \exists x\phi \to {\exists }^{*}x\phi$$

Proof

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