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Mirrors > Home > ILE Home > Th. List > nfned | GIF version |
Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfned.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfned.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfned | ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2328 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | nfned.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfned.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfeqd 2314 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
5 | 4 | nfnd 1637 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵) |
6 | 1, 5 | nfxfrd 1455 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1335 Ⅎwnf 1440 Ⅎwnfc 2286 ≠ wne 2327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-cleq 2150 df-nfc 2288 df-ne 2328 |
This theorem is referenced by: (None) |
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