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Mirrors > Home > ILE Home > Th. List > dvelimc | GIF version |
Description: Version of dvelim 1997 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
dvelimc.1 | ⊢ Ⅎ𝑥𝐴 |
dvelimc.2 | ⊢ Ⅎ𝑧𝐵 |
dvelimc.3 | ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
dvelimc | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1446 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nftru 1446 | . . 3 ⊢ Ⅎ𝑧⊤ | |
3 | dvelimc.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
5 | dvelimc.2 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
6 | 5 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑧𝐵) |
7 | dvelimc.3 | . . . 4 ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) | |
8 | 7 | a1i 9 | . . 3 ⊢ (⊤ → (𝑧 = 𝑦 → 𝐴 = 𝐵)) |
9 | 1, 2, 4, 6, 8 | dvelimdc 2320 | . 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
10 | 9 | mptru 1344 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1333 = wceq 1335 ⊤wtru 1336 Ⅎwnfc 2286 |
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-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 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-sb 1743 df-cleq 2150 df-clel 2153 df-nfc 2288 |
This theorem is referenced by: nfcvf 2322 |
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