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Theorem hbmo 1981
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 1946 . 2 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2 hbmo.1 . . . 4 (𝜑 → ∀𝑥𝜑)
32hbex 1568 . . 3 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
42hbeu 1963 . . 3 (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
53, 4hbim 1478 . 2 ((∃𝑦𝜑 → ∃!𝑦𝜑) → ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
61, 5hbxfrbi 1402 1 (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wex 1422  ∃!weu 1942  ∃*wmo 1943
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by:  moexexdc  2026  2moex  2028  2euex  2029  2exeu  2034
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