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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 2341 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | nfned.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfned.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfeqd 2327 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
5 | 4 | nfnd 1650 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵) |
6 | 1, 5 | nfxfrd 1468 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 Ⅎwnf 1453 Ⅎwnfc 2299 ≠ wne 2340 |
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 609 ax-in2 610 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-cleq 2163 df-nfc 2301 df-ne 2341 |
This theorem is referenced by: (None) |
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