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Theorem hbmo 1987
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 1952 . 2 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2 hbmo.1 . . . 4 (𝜑 → ∀𝑥𝜑)
32hbex 1572 . . 3 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
42hbeu 1969 . . 3 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
53, 4hbim 1482 . 2 ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
61, 5hbxfrbi 1406 1 (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wex 1426  ∃!weu 1948  ∃*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  moexexdc  2032  2moex  2034  2euex  2035  2exeu  2040
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