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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 2023 | . 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
2 | hbmo.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbex 1629 | . . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
4 | 2 | hbeu 2040 | . . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) |
5 | 3, 4 | hbim 1538 | . 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) |
6 | 1, 5 | hbxfrbi 1465 | 1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 ∃wex 1485 ∃!weu 2019 ∃*wmo 2020 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 |
This theorem is referenced by: moexexdc 2103 2moex 2105 2euex 2106 2exeu 2111 |
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