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Theorem hbmo 2077
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 2042 . 2 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2 hbmo.1 . . . 4 (𝜑 → ∀𝑥𝜑)
32hbex 1647 . . 3 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
42hbeu 2059 . . 3 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
53, 4hbim 1556 . 2 ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
61, 5hbxfrbi 1483 1 (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wex 1503  ∃!weu 2038  ∃*wmo 2039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042
This theorem is referenced by:  moexexdc  2122  2moex  2124  2euex  2125  2exeu  2130
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