Theorem nfned 2454

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

Hypotheses

Ref	Expression
nfned.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfned.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfned	⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)

Proof of Theorem nfned

Step	Hyp	Ref	Expression
1		df-ne 2361	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		nfned.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfned.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
4	2, 3	nfeqd 2347	. . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
5	4	nfnd 1668	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵)
6	1, 5	nfxfrd 1486	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364  Ⅎwnf 1471  Ⅎwnfc 2319   ≠ wne 2360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-cleq 2182  df-nfc 2321  df-ne 2361

This theorem is referenced by: (None)