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Mirrors > Home > ILE Home > Th. List > nfeu | GIF version |
Description: Bound-variable hypothesis builder for existential uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.) |
Ref | Expression |
---|---|
nfeu.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfeu | ⊢ Ⅎ𝑥∃!𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb8eu 2019 | . 2 ⊢ (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑) |
3 | nfeu.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | 3 | nfsb 1926 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜑 |
5 | 4 | nfeuv 2024 | . 2 ⊢ Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜑 |
6 | 2, 5 | nfxfr 1454 | 1 ⊢ Ⅎ𝑥∃!𝑦𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1440 [wsb 1742 ∃!weu 2006 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 |
This theorem is referenced by: hbeu 2027 eusv2nf 4417 |
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