Theorem nfdm

Description: Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004) (Revised by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypothesis	nfrn.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	nfdm	$$⊢\underset{\_}{Ⅎ}x\mathrm{dom}A$$

Proof

Step	Hyp	Ref	Expression
1		nfrn.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		df-dm	$$⊢\mathrm{dom}A=\left\{y|\exists zyAz\right\}$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xy$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xz$$
5	3 1 4	nfbr	$$⊢ℲxyAz$$
6	5	nfex	$$⊢Ⅎx\exists zyAz$$
7	6	nfab	$$⊢\underset{\_}{Ⅎ}x\left\{y|\exists zyAz\right\}$$
8	2 7	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\mathrm{dom}A$$