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Theorem eumo 2051
Description: Existential uniqueness implies "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
Assertion
Ref Expression
eumo (∃!𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem eumo
StepHypRef Expression
1 ax-1 6 . 2 (∃!𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2 df-mo 2023 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
31, 2sylibr 133 1 (∃!𝑥𝜑 → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1485  ∃!weu 2019  ∃*wmo 2020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-mo 2023
This theorem is referenced by:  eumoi  2052  eu5  2066  euimmo  2086  moaneu  2095  eupick  2098  2eumo  2107  moeq3dc  2906  nfunsn  5528  fnoprabg  5952  uptx  13033  txcn  13034
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