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Mirrors > Home > MPE Home > Th. List > dvelimc | Structured version Visualization version GIF version |
Description: Version of dvelim 2450 for classes. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 1812 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nftru 1812 | . . 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 2924 | . 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
10 | 9 | mptru 1550 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 = wceq 1543 ⊤wtru 1544 Ⅎwnfc 2877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-13 2371 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-cleq 2728 df-clel 2809 df-nfc 2879 |
This theorem is referenced by: (None) |
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