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	⊢ Ⅎ 𝑥 𝐴
	nfne.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfne	⊢ Ⅎ 𝑥 𝐴 ≠ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfne.1	⊢ Ⅎ 𝑥 𝐴
2		nfne.2	⊢ Ⅎ 𝑥 𝐵
3		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
4	1 2	nfeq	⊢ Ⅎ 𝑥 𝐴 = 𝐵
5	4	nfn	⊢ Ⅎ 𝑥 ¬ 𝐴 = 𝐵
6	3 5	nfxfr	⊢ Ⅎ 𝑥 𝐴 ≠ 𝐵