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Theorem mo2r 2007
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 1464 . . . 4 (𝜑 → ∀𝑦𝜑)
32eu3h 2000 . . 3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
43simplbi2com 1385 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
5 df-mo 1959 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
64, 5sylibr 133 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1294  wnf 1401  wex 1433  ∃!weu 1955  ∃*wmo 1956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959
This theorem is referenced by:  mo2icl  2808  rmo2ilem  2942  dffun5r  5061  frecuzrdgtcl  9968  frecuzrdgfunlem  9975
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