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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 1511 | . . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → 𝑦 = 𝐴) | |
2 | sp 1511 | . . . 4 ⊢ (∀𝑥 𝑧 = 𝐴 → 𝑧 = 𝐴) | |
3 | eqtr3 2197 | . . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → 𝑦 = 𝑧) | |
4 | 1, 2, 3 | syl2an 289 | . . 3 ⊢ ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧) |
5 | 4 | gen2 1450 | . 2 ⊢ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧) |
6 | eqeq1 2184 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐴 ↔ 𝑧 = 𝐴)) | |
7 | 6 | albidv 1824 | . . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴)) |
8 | 7 | eu4 2088 | . 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (∃𝑦∀𝑥 𝑦 = 𝐴 ∧ ∀𝑦∀𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧))) |
9 | 5, 8 | mpbiran2 941 | 1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥 𝑦 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 = wceq 1353 ∃wex 1492 ∃!weu 2026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-cleq 2170 |
This theorem is referenced by: eusvnfb 4454 |
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