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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 3539 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | vex 2644 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | snex 4049 | . . . 4 ⊢ {𝑦} ∈ V |
4 | eleq1 2162 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} ∈ V ↔ {𝑦} ∈ V)) | |
5 | 3, 4 | mpbiri 167 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
6 | 5 | exlimiv 1545 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
7 | 1, 6 | sylbi 120 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∃wex 1436 ∈ wcel 1448 ∃!weu 1960 {cab 2086 Vcvv 2641 {csn 3474 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 |
This theorem is referenced by: (None) |
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