Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfeuv | GIF version |
Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1919 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 1421 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
3 | 1, 2 | nfbi 1481 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ↔ 𝑦 = 𝑧) |
4 | 3 | nfal 1468 | . . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 ↔ 𝑦 = 𝑧) |
5 | 4 | nfex 1528 | . 2 ⊢ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧) |
6 | df-eu 1903 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)) | |
7 | 6 | nfbii 1362 | . 2 ⊢ (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)) |
8 | 5, 7 | mpbir 134 | 1 ⊢ Ⅎ𝑥∃!𝑦𝜑 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1241 Ⅎwnf 1349 ∃wex 1381 ∃!weu 1900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-eu 1903 |
This theorem is referenced by: nfeu 1919 |
Copyright terms: Public domain | W3C validator |