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Mirrors > Home > ILE Home > Th. List > euabex | GIF version |
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
Ref | Expression |
---|---|
euabex | ⊢ (∃!xφ → {x ∣ φ} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 3430 | . 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y}) | |
2 | vex 2554 | . . . . 5 ⊢ y ∈ V | |
3 | snexgOLD 3926 | . . . . 5 ⊢ (y ∈ V → {y} ∈ V) | |
4 | 2, 3 | ax-mp 7 | . . . 4 ⊢ {y} ∈ V |
5 | eleq1 2097 | . . . 4 ⊢ ({x ∣ φ} = {y} → ({x ∣ φ} ∈ V ↔ {y} ∈ V)) | |
6 | 4, 5 | mpbiri 157 | . . 3 ⊢ ({x ∣ φ} = {y} → {x ∣ φ} ∈ V) |
7 | 6 | exlimiv 1486 | . 2 ⊢ (∃y{x ∣ φ} = {y} → {x ∣ φ} ∈ V) |
8 | 1, 7 | sylbi 114 | 1 ⊢ (∃!xφ → {x ∣ φ} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 {cab 2023 Vcvv 2551 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: (None) |
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