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Mirrors > Home > ILE Home > Th. List > eunex | GIF version |
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
eunex | ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | eu3 1987 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
3 | dtruex 4302 | . . . . 5 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | |
4 | nfa1 1474 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑦) | |
5 | sp 1441 | . . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | |
6 | 5 | con3d 593 | . . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑)) |
7 | 4, 6 | eximd 1543 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑)) |
8 | 3, 7 | mpi 15 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑) |
9 | 8 | exlimiv 1529 | . . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑) |
10 | 9 | adantl 271 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑) |
11 | 2, 10 | sylbi 119 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∀wal 1282 = wceq 1284 ∃wex 1421 ∃!weu 1941 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 |
This theorem is referenced by: (None) |
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