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Theorem eumo 2029
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 2001 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
31, 2sylibr 133 1 (∃!𝑥𝜑 → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1468  ∃!weu 1997  ∃*wmo 1998
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 2001
This theorem is referenced by:  eumoi  2030  eu5  2044  euimmo  2064  moaneu  2073  eupick  2076  2eumo  2085  moeq3dc  2855  nfunsn  5448  fnoprabg  5865  uptx  12432  txcn  12433
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