![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eusvnfb | GIF version |
Description: Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
eusvnfb | ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eusvnf 4288 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴) | |
2 | euex 1979 | . . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃𝑦∀𝑥 𝑦 = 𝐴) | |
3 | id 19 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
4 | vex 2623 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | syl6eqelr 2180 | . . . . . 6 ⊢ (𝑦 = 𝐴 → 𝐴 ∈ V) |
6 | 5 | sps 1476 | . . . . 5 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝐴 ∈ V) |
7 | 6 | exlimiv 1535 | . . . 4 ⊢ (∃𝑦∀𝑥 𝑦 = 𝐴 → 𝐴 ∈ V) |
8 | 2, 7 | syl 14 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → 𝐴 ∈ V) |
9 | 1, 8 | jca 301 | . 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) |
10 | isset 2626 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
11 | nfcvd 2230 | . . . . . . . 8 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
12 | id 19 | . . . . . . . 8 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
13 | 11, 12 | nfeqd 2244 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
14 | 13 | nfrd 1459 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → (𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴)) |
15 | 14 | eximdv 1809 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 → ∃𝑦∀𝑥 𝑦 = 𝐴)) |
16 | 10, 15 | syl5bi 151 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V → ∃𝑦∀𝑥 𝑦 = 𝐴)) |
17 | 16 | imp 123 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V) → ∃𝑦∀𝑥 𝑦 = 𝐴) |
18 | eusv1 4287 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴) | |
19 | 17, 18 | sylibr 133 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V) → ∃!𝑦∀𝑥 𝑦 = 𝐴) |
20 | 9, 19 | impbii 125 | 1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1288 = wceq 1290 ∃wex 1427 ∈ wcel 1439 ∃!weu 1949 Ⅎwnfc 2216 Vcvv 2620 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-sbc 2842 df-csb 2935 |
This theorem is referenced by: eusv2nf 4291 eusv2 4292 |
Copyright terms: Public domain | W3C validator |