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Theorem eu5 2073
Description: Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
Assertion
Ref Expression
eu5 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Proof of Theorem eu5
StepHypRef Expression
1 euex 2056 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
2 eumo 2058 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
31, 2jca 306 . 2 (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
4 df-mo 2030 . . . . 5 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
54biimpi 120 . . . 4 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
65imp 124 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥𝜑) → ∃!𝑥𝜑)
76ancoms 268 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) → ∃!𝑥𝜑)
83, 7impbii 126 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1492  ∃!weu 2026  ∃*wmo 2027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030
This theorem is referenced by:  exmoeu2  2074  euan  2082  eu4  2088  euim  2094  euexex  2111  2euex  2113  2euswapdc  2117  2exeu  2118  reu5  2690  reuss2  3416  funcnv3  5279  fnres  5333  fnopabg  5340  brprcneu  5509  dff3im  5662  recmulnqg  7390  uptx  13777
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