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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 1461 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb8eu 1954 | . 2 ⊢ (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑) |
3 | nfeu.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | 3 | nfsb 1863 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜑 |
5 | 4 | nfeuv 1959 | . 2 ⊢ Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜑 |
6 | 2, 5 | nfxfr 1403 | 1 ⊢ Ⅎ𝑥∃!𝑦𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1389 [wsb 1685 ∃!weu 1941 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 |
This theorem is referenced by: hbeu 1962 eusv2nf 4206 |
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