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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 1539 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | eu3 2084 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
3 | dtruex 4576 | . . . . 5 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | |
4 | nfa1 1552 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑦) | |
5 | sp 1522 | . . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)) | |
6 | 5 | con3d 632 | . . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑)) |
7 | 4, 6 | eximd 1623 | . . . . 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 1609 | . . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑) |
10 | 9 | adantl 277 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑) |
11 | 2, 10 | sylbi 121 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∀wal 1362 = wceq 1364 ∃wex 1503 ∃!weu 2038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 |
This theorem is referenced by: (None) |
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