Theorem nfmod2

Description: Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 . See nfmodv for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016) Avoid df-eu . (Revised by BJ, 14-Oct-2022) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfmod2.1	$$⊢Ⅎy\phi$$
	nfmod2.2	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
Assertion	nfmod2	$$⊢\phi \to Ⅎx{\exists }^{*}y\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfmod2.1	$$⊢Ⅎy\phi$$
2		nfmod2.2	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
3		df-mo	$$⊢{\exists }^{*}y\psi \leftrightarrow \exists z\forall y\left(\psi \to y=z\right)$$
4		nfv	$$⊢Ⅎz\phi$$
5		nfeqf1	$$⊢\neg \forall xx=y\to Ⅎxy=z$$
6	5	adantl	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎxy=z$$
7	2 6	nfimd	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\left(\psi \to y=z\right)$$
8	1 7	nfald2	$$⊢\phi \to Ⅎx\forall y\left(\psi \to y=z\right)$$
9	4 8	nfexd	$$⊢\phi \to Ⅎx\exists z\forall y\left(\psi \to y=z\right)$$
10	3 9	nfxfrd	$$⊢\phi \to Ⅎx{\exists }^{*}y\psi$$