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Mirrors > Home > ILE Home > Th. List > dvelimc | GIF version |
Description: Version of dvelim 1893 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 1355 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nftru 1355 | . . 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 2197 | . 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
10 | 9 | trud 1252 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1241 = wceq 1243 ⊤wtru 1244 Ⅎwnfc 2165 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 |
This theorem is referenced by: nfcvf 2199 |
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