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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 2004 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | hbe1 1472 | . . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
3 | hbeu1 2010 | . . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑) | |
4 | 2, 3 | hbim 1525 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | 1, 4 | hbxfrbi 1449 | 1 ⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 ∃wex 1469 ∃!weu 2000 ∃*wmo 2001 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-eu 2003 df-mo 2004 |
This theorem is referenced by: mopick2 2083 moexexdc 2084 |
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