Theorem nfnel

Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011) (Revised by Mario Carneiro, 7-Oct-2016)

	Ref	Expression
Hypotheses	nfnel.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfnel.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfnel	$$⊢ℲxA\notin B$$

Proof

Step	Hyp	Ref	Expression
1		nfnel.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfnel.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		df-nel	$$⊢A\notin B\leftrightarrow \neg A\in B$$
4	1 2	nfel	$$⊢ℲxA\in B$$
5	4	nfn	$$⊢Ⅎx\neg A\in B$$
6	3 5	nfxfr	$$⊢ℲxA\notin B$$