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Mirrors > Home > ILE Home > Th. List > eu3 | GIF version |
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
Ref | Expression |
---|---|
eu3.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
eu3 | ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu3.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1499 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | eu3h 2044 | 1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 Ⅎwnf 1436 ∃wex 1468 ∃!weu 1999 |
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 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2002 |
This theorem is referenced by: eqeu 2854 reu3 2874 eunex 4476 |
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