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Mirrors > Home > ILE Home > Th. List > hbmo1 | GIF version |
Description: Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) |
Ref | Expression |
---|---|
hbmo1 | ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 1904 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | hbe1 1384 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
3 | hbeu1 1910 | . . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) | |
4 | 2, 3 | hbim 1437 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | 1, 4 | hbxfrbi 1361 | 1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑) |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-eu 1903 df-mo 1904 |
This theorem is referenced by: mopick2 1983 moexexdc 1984 |
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