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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 1466 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | eu3 1994 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
3 | dtruex 4375 | . . . . 5 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | |
4 | nfa1 1479 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑦) | |
5 | sp 1446 | . . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | |
6 | 5 | con3d 596 | . . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑)) |
7 | 4, 6 | eximd 1548 | . . . . 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 1534 | . . 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 1287 = wceq 1289 ∃wex 1426 ∃!weu 1948 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-setind 4353 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-v 2621 df-dif 3001 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 |
This theorem is referenced by: (None) |
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