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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 2042 | . 2 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
2 | hbmo.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbex 1647 | . . 3 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) |
4 | 2 | hbeu 2059 | . . 3 ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑) |
5 | 3, 4 | hbim 1556 | . 2 ⊢ ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) |
6 | 1, 5 | hbxfrbi 1483 | 1 ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 ∃wex 1503 ∃!weu 2038 ∃*wmo 2039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 |
This theorem is referenced by: moexexdc 2122 2moex 2124 2euex 2125 2exeu 2130 |
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