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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 2010 | . 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
2 | hbmo.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbex 1616 | . . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
4 | 2 | hbeu 2027 | . . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) |
5 | 3, 4 | hbim 1525 | . 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) |
6 | 1, 5 | hbxfrbi 1452 | 1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 ∃wex 1472 ∃!weu 2006 ∃*wmo 2007 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 |
This theorem is referenced by: moexexdc 2090 2moex 2092 2euex 2093 2exeu 2098 |
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