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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-euequf | Structured version Visualization version GIF version |
Description: euequ 2669 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.) |
Ref | Expression |
---|---|
wl-euequf | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6ev 2077 | . . 3 ⊢ ∃𝑧 𝑧 = 𝑦 | |
2 | nfv 2013 | . . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | |
3 | nfna1 2203 | . . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
4 | nfeqf2 2394 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
5 | equequ2 2130 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) | |
6 | 5 | equcoms 2124 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) |
7 | 6 | a1i 11 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))) |
8 | 3, 4, 7 | alrimdd 2257 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))) |
9 | 2, 8 | eximd 2259 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))) |
10 | 1, 9 | mpi 20 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) |
11 | eu6 2645 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) | |
12 | 10, 11 | sylibr 226 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1654 ∃wex 1878 ∃!weu 2639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ex 1879 df-nf 1883 df-mo 2605 df-eu 2640 |
This theorem is referenced by: (None) |
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