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Mirrors > Home > ILE Home > Th. List > nfeuv | GIF version |
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2019 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.) |
Ref | Expression |
---|---|
nfeuv.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfeuv | ⊢ Ⅎ𝑥∃!𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeuv.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1509 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
3 | 1, 2 | nfbi 1569 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ↔ 𝑦 = 𝑧) |
4 | 3 | nfal 1556 | . . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 ↔ 𝑦 = 𝑧) |
5 | 4 | nfex 1617 | . 2 ⊢ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧) |
6 | df-eu 2003 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)) | |
7 | 6 | nfbii 1450 | . 2 ⊢ (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)) |
8 | 5, 7 | mpbir 145 | 1 ⊢ Ⅎ𝑥∃!𝑦𝜑 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1330 Ⅎwnf 1437 ∃wex 1469 ∃!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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-eu 2003 |
This theorem is referenced by: nfeu 2019 |
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