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Mirrors > Home > ILE Home > Th. List > hbeu | GIF version |
Description: Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.) |
Ref | Expression |
---|---|
hbeu.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbeu | ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbeu.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nfi 1438 | . . 3 ⊢ Ⅎ𝑥𝜑 |
3 | 2 | nfeu 2016 | . 2 ⊢ Ⅎ𝑥∃!𝑦𝜑 |
4 | 3 | nfri 1499 | 1 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 ∃!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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 |
This theorem is referenced by: hbmo 2036 2eu7 2091 |
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