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Theorem hbmo1 1983
Description: Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)
Assertion
Ref Expression
hbmo1 (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)

Proof of Theorem hbmo1
StepHypRef Expression
1 df-mo 1949 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 hbe1 1427 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
3 hbeu1 1955 . . 3 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
42, 3hbim 1480 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑))
51, 4hbxfrbi 1404 1 (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  wex 1424  ∃!weu 1945  ∃*wmo 1946
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-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-eu 1948  df-mo 1949
This theorem is referenced by:  mopick2  2028  moexexdc  2029
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