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Mirrors > Home > ILE Home > Th. List > nfeudv | GIF version |
Description: Deduction version of nfeu 2057. Similar to nfeud 2054 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 1539 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfeudv.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeudv.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | nfv 1539 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
5 | 4 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) |
6 | 3, 5 | nfbid 1599 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
7 | 2, 6 | nfald 1771 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
8 | 1, 7 | nfexd 1772 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
9 | df-eu 2041 | . . 3 ⊢ (∃!𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) | |
10 | 9 | nfbii 1484 | . 2 ⊢ (Ⅎ𝑥∃!𝑦𝜓 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
11 | 8, 10 | sylibr 134 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 = wceq 1364 Ⅎwnf 1471 ∃wex 1503 ∃!weu 2038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-eu 2041 |
This theorem is referenced by: nfeud 2054 |
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