Theorem nfned

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

	Ref	Expression
Hypotheses	nfned.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
	nfned.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion	nfned	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfned.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
2		nfned.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
3		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
4	1 2	nfeqd	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 = 𝐵 )
5	4	nfnd	⊢ ( 𝜑 → Ⅎ 𝑥 ¬ 𝐴 = 𝐵 )
6	3 5	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ≠ 𝐵 )