![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dvelimc | Structured version Visualization version GIF version |
Description: Version of dvelim 2442 for classes. Usage of this theorem is discouraged because it depends on ax-13 2363. (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 1798 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nftru 1798 | . . 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 2922 | . 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
10 | 9 | mptru 1540 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 = wceq 1533 ⊤wtru 1534 Ⅎwnfc 2875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2363 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-cleq 2716 df-clel 2802 df-nfc 2877 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |