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Mirrors > Home > ILE Home > Th. List > euabex | Structured version 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 3412 | . 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y}) | |
2 | vex 2537 | . . . . 5 ⊢ y ∈ V | |
3 | snexgOLD 3908 | . . . . 5 ⊢ (y ∈ V → {y} ∈ V) | |
4 | 2, 3 | ax-mp 7 | . . . 4 ⊢ {y} ∈ V |
5 | eleq1 2083 | . . . 4 ⊢ ({x ∣ φ} = {y} → ({x ∣ φ} ∈ V ↔ {y} ∈ V)) | |
6 | 4, 5 | mpbiri 157 | . . 3 ⊢ ({x ∣ φ} = {y} → {x ∣ φ} ∈ V) |
7 | 6 | exlimiv 1473 | . 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 1228 ∃wex 1363 ∈ wcel 1375 ∃!weu 1883 {cab 2009 Vcvv 2534 {csn 3349 |
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 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-10 1378 ax-11 1379 ax-i12 1380 ax-bnd 1381 ax-4 1382 ax-14 1387 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-i5r 1411 ax-ext 2005 ax-sep 3848 ax-pow 3900 |
This theorem depends on definitions: df-bi 110 df-tru 1231 df-nf 1330 df-sb 1629 df-eu 1886 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-v 2536 df-in 2900 df-ss 2907 df-pw 3335 df-sn 3355 |
This theorem is referenced by: (None) |
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