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Mirrors > Home > ILE Home > Th. List > hbmo | GIF version |
Description: Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Ref | Expression |
---|---|
hbmo.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbmo | ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 1904 | . 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
2 | hbmo.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbex 1527 | . . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
4 | 2 | hbeu 1921 | . . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) |
5 | 3, 4 | hbim 1437 | . 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) |
6 | 1, 5 | hbxfrbi 1361 | 1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 ∃wex 1381 ∃!weu 1900 ∃*wmo 1901 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 |
This theorem is referenced by: moexexdc 1984 2moex 1986 2euex 1987 2exeu 1992 |
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