Theorem nfne

Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007) (Revised by Mario Carneiro, 7-Oct-2016)

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

Proof

Step	Hyp	Ref	Expression
1		nfne.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfne.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
4	1 2	nfeq	$$⊢ℲxA=B$$
5	4	nfn	$$⊢Ⅎx\neg A=B$$
6	3 5	nfxfr	$$⊢ℲxA\ne B$$