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Theorem mo2r 2051
 Description: A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
Hypothesis
Ref Expression
mo2r.1 𝑦𝜑
Assertion
Ref Expression
mo2r (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2r
StepHypRef Expression
1 mo2r.1 . . . . 5 𝑦𝜑
21nfri 1499 . . . 4 (𝜑 → ∀𝑦𝜑)
32eu3h 2044 . . 3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
43simplbi2com 1420 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
5 df-mo 2003 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
64, 5sylibr 133 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468  ∃!weu 1999  ∃*wmo 2000 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003 This theorem is referenced by:  mo2icl  2863  rmo2ilem  2998  dffun5r  5135  frecuzrdgtcl  10185  frecuzrdgfunlem  10192
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