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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 1981 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | hbe1 1456 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
3 | hbeu1 1987 | . . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) | |
4 | 2, 3 | hbim 1509 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | 1, 4 | hbxfrbi 1433 | 1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1314 ∃wex 1453 ∃!weu 1977 ∃*wmo 1978 |
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-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-eu 1980 df-mo 1981 |
This theorem is referenced by: mopick2 2060 moexexdc 2061 |
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