Theorem nfne 2495

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

Hypotheses

Ref	Expression
nfne.1	⊢ Ⅎ𝑥𝐴
nfne.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfne	⊢ Ⅎ𝑥 𝐴 ≠ 𝐵

Proof of Theorem nfne

Step	Hyp	Ref	Expression
1		df-ne 2403	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		nfne.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfne.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfeq 2382	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
5	4	nfn 1706	. 2 ⊢ Ⅎ𝑥 ¬ 𝐴 = 𝐵
6	1, 5	nfxfr 1522	1 ⊢ Ⅎ𝑥 𝐴 ≠ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1397  Ⅎwnf 1508  Ⅎwnfc 2361   ≠ wne 2402

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403

This theorem is referenced by: (None)