Theorem abidnf

Description: Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005) (Proof shortened by Mario Carneiro, 12-Oct-2016)

	Ref	Expression
Assertion	abidnf	$$⊢\underset{\_}{Ⅎ}xA\to \left\{z|\forall xz\in A\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		sp	$$⊢\forall xz\in A\to z\in A$$
2		nfcr	$$⊢\underset{\_}{Ⅎ}xA\to Ⅎxz\in A$$
3	2	nf5rd	$$⊢\underset{\_}{Ⅎ}xA\to \left(z\in A\to \forall xz\in A\right)$$
4	1 3	impbid2	$$⊢\underset{\_}{Ⅎ}xA\to \left(\forall xz\in A\leftrightarrow z\in A\right)$$
5	4	eqabcdv	$$⊢\underset{\_}{Ⅎ}xA\to \left\{z|\forall xz\in A\right\}=A$$