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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 3485 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | vex 2615 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | snex 3983 | . . . 4 ⊢ {𝑦} ∈ V |
4 | eleq1 2145 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} ∈ V ↔ {𝑦} ∈ V)) | |
5 | 3, 4 | mpbiri 166 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
6 | 5 | exlimiv 1530 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
7 | 1, 6 | sylbi 119 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∃!weu 1943 {cab 2069 Vcvv 2612 {csn 3422 |
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-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 |
This theorem is referenced by: (None) |
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