Theorem nfeud2

Description: Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . Check out nfeudw for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016) (Proof shortened by Wolf Lammen, 4-Oct-2018) (Proof shortened by BJ, 14-Oct-2022) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfeud2.1	$$⊢Ⅎy\phi$$
	nfeud2.2	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
Assertion	nfeud2	$$⊢\phi \to Ⅎx\exists !y\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfeud2.1	$$⊢Ⅎy\phi$$
2		nfeud2.2	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
3		df-eu	$$⊢\exists !y\psi \leftrightarrow \left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
4	1 2	nfexd2	$$⊢\phi \to Ⅎx\exists y\psi$$
5	1 2	nfmod2	$$⊢\phi \to Ⅎx{\exists }^{*}y\psi$$
6	4 5	nfand	$$⊢\phi \to Ⅎx\left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
7	3 6	nfxfrd	$$⊢\phi \to Ⅎx\exists !y\psi$$