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Mirrors > Home > ILE Home > Th. List > nfeudv | GIF version |
Description: Deduction version of nfeu 2016. Similar to nfeud 2013 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.) |
Ref | Expression |
---|---|
nfeudv.1 | ⊢ Ⅎ𝑦𝜑 |
nfeudv.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfeudv | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfeudv.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeudv.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
5 | 4 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) |
6 | 3, 5 | nfbid 1567 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
7 | 2, 6 | nfald 1733 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
8 | 1, 7 | nfexd 1734 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
9 | df-eu 2000 | . . 3 ⊢ (∃!𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) | |
10 | 9 | nfbii 1449 | . 2 ⊢ (Ⅎ𝑥∃!𝑦𝜓 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
11 | 8, 10 | sylibr 133 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 ∃!weu 1997 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-eu 2000 |
This theorem is referenced by: nfeud 2013 |
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