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Theorem hbmo1 1938
 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 1904 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 hbe1 1384 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
3 hbeu1 1910 . . 3 (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
42, 3hbim 1437 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∀𝑥(∃𝑥𝜑 → ∃!𝑥𝜑))
51, 4hbxfrbi 1361 1 (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-eu 1903  df-mo 1904 This theorem is referenced by:  mopick2  1983  moexexdc  1984
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