Theorem nfeudw

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu . Version of nfeud with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 15-Feb-2013) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfeudw.1	$$⊢Ⅎy\phi$$
	nfeudw.2	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfeudw	$$⊢\phi \to Ⅎx\exists !y\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfeudw.1	$$⊢Ⅎy\phi$$
2		nfeudw.2	$$⊢\phi \to Ⅎx\psi$$
3		df-eu	$$⊢\exists !y\psi \leftrightarrow \left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
4	1 2	nfexd	$$⊢\phi \to Ⅎx\exists y\psi$$
5	1 2	nfmodv	$$⊢\phi \to Ⅎx{\exists }^{*}y\psi$$
6	4 5	nfand	$$⊢\phi \to Ⅎx\left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
7	3 6	nfxfrd	$$⊢\phi \to Ⅎx\exists !y\psi$$