Theorem nfmodv

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod for a version without disjoint variable conditions but requiring ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016) (Revised by BJ, 28-Jan-2023)

	Ref	Expression
Hypotheses	nfmodv.1	$$⊢Ⅎy\phi$$
	nfmodv.2	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfmodv	$$⊢\phi \to Ⅎx{\exists }^{*}y\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfmodv.1	$$⊢Ⅎy\phi$$
2		nfmodv.2	$$⊢\phi \to Ⅎx\psi$$
3		dfmo	$$⊢{\exists }^{*}y\psi \leftrightarrow \exists z\forall y\left(\psi \to y=z\right)$$
4		nfv	$$⊢Ⅎz\phi$$
5		nfvd	$$⊢\phi \to Ⅎxy=z$$
6	2 5	nfimd	$$⊢\phi \to Ⅎx\left(\psi \to y=z\right)$$
7	1 6	nfald	$$⊢\phi \to Ⅎx\forall y\left(\psi \to y=z\right)$$
8	4 7	nfexd	$$⊢\phi \to Ⅎx\exists z\forall y\left(\psi \to y=z\right)$$
9	3 8	nfxfrd	$$⊢\phi \to Ⅎx{\exists }^{*}y\psi$$