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| Mirrors > Home > MPE Home > Th. List > dvelimc | Structured version Visualization version GIF version | ||
| Description: Version of dvelim 2455 for classes. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) | 
| 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 1803 | . . 3 ⊢ Ⅎ𝑥⊤ | |
| 2 | nftru 1803 | . . 3 ⊢ Ⅎ𝑧⊤ | |
| 3 | dvelimc.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) | 
| 5 | dvelimc.2 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
| 6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑧𝐵) | 
| 7 | dvelimc.3 | . . . 4 ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵) | |
| 8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → (𝑧 = 𝑦 → 𝐴 = 𝐵)) | 
| 9 | 1, 2, 4, 6, 8 | dvelimdc 2929 | . 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) | 
| 10 | 9 | mptru 1546 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 = wceq 1539 ⊤wtru 1540 Ⅎwnfc 2889 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-cleq 2728 df-clel 2815 df-nfc 2891 | 
| This theorem is referenced by: (None) | 
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