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Mirrors > Home > ILE Home > Th. List > eusv1 | GIF version |
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) |
Ref | Expression |
---|---|
eusv1 | ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 1488 | . . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴) | |
2 | sp 1488 | . . . 4 ⊢ (∀𝑥 𝑧 = 𝐴 → 𝑧 = 𝐴) | |
3 | eqtr3 2157 | . . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → 𝑦 = 𝑧) | |
4 | 1, 2, 3 | syl2an 287 | . . 3 ⊢ ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧) |
5 | 4 | gen2 1426 | . 2 ⊢ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧) |
6 | eqeq1 2144 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑧 = 𝐴)) | |
7 | 6 | albidv 1796 | . . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴)) |
8 | 7 | eu4 2059 | . 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (∃𝑦∀𝑥 𝑦 = 𝐴 ∧ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧))) |
9 | 5, 8 | mpbiran2 925 | 1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 = wceq 1331 ∃wex 1468 ∃!weu 1997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-cleq 2130 |
This theorem is referenced by: eusvnfb 4370 |
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