Theorem nfneld

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	nfneld.1	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	nfneld.2	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
Assertion	nfneld	$$⊢\phi \to ℲxA\notin B$$

Proof

Step	Hyp	Ref	Expression
1		nfneld.1	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
2		nfneld.2	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
3		df-nel	$$⊢A\notin B\leftrightarrow \neg A\in B$$
4	1 2	nfeld	$$⊢\phi \to ℲxA\in B$$
5	4	nfnd	$$⊢\phi \to Ⅎx\neg A\in B$$
6	3 5	nfxfrd	$$⊢\phi \to ℲxA\notin B$$