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Theorem hbmo 2045
Description: Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
Hypothesis
Ref Expression
hbmo.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbmo (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)

Proof of Theorem hbmo
StepHypRef Expression
1 df-mo 2010 . 2 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2 hbmo.1 . . . 4 (𝜑 → ∀𝑥𝜑)
32hbex 1616 . . 3 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
42hbeu 2027 . . 3 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
53, 4hbim 1525 . 2 ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
61, 5hbxfrbi 1452 1 (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wex 1472  ∃!weu 2006  ∃*wmo 2007
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010
This theorem is referenced by:  moexexdc  2090  2moex  2092  2euex  2093  2exeu  2098
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