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Theorem hbmo 1939
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 1904 . 2 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2 hbmo.1 . . . 4 (𝜑 → ∀𝑥𝜑)
32hbex 1527 . . 3 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
42hbeu 1921 . . 3 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
53, 4hbim 1437 . 2 ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
61, 5hbxfrbi 1361 1 (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by:  moexexdc  1984  2moex  1986  2euex  1987  2exeu  1992
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