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Theorem eu5 1995
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 1978 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
2 eumo 1980 . . 3 (∃!𝑥𝜑 → ∃*𝑥𝜑)
31, 2jca 300 . 2 (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
4 df-mo 1952 . . . . 5 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
54biimpi 118 . . . 4 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
65imp 122 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥𝜑) → ∃!𝑥𝜑)
76ancoms 264 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) → ∃!𝑥𝜑)
83, 7impbii 124 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1426  ∃!weu 1948  ∃*wmo 1949
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  exmoeu2  1996  euan  2004  eu4  2010  euim  2016  euexex  2033  2euex  2035  2euswapdc  2039  2exeu  2040  reu5  2579  reuss2  3277  funcnv3  5062  fnres  5116  fnopabg  5123  brprcneu  5282  dff3im  5428  recmulnqg  6929
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