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Mirrors > Home > ILE Home > Th. List > nfeud | GIF version |
Description: Deduction version of nfeu 2019. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
Ref | Expression |
---|---|
nfeud.1 | ⊢ Ⅎ𝑦𝜑 |
nfeud.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfeud | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 | . . 3 ⊢ Ⅎ𝑧𝜓 | |
2 | 1 | sb8eu 2013 | . 2 ⊢ (∃!𝑦𝜓 ↔ ∃!𝑧[𝑧 / 𝑦]𝜓) |
3 | nfv 1509 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | nfeud.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
5 | nfeud.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfsbd 1951 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
7 | 3, 6 | nfeudv 2015 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜓) |
8 | 2, 7 | nfxfrd 1452 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1437 [wsb 1736 ∃!weu 2000 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 |
This theorem is referenced by: nfmod 2017 hbeud 2022 nfreudxy 2607 |
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