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Theorem mo2r 1995
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 1453 . . . 4 (𝜑 → ∀𝑦𝜑)
32eu3h 1988 . . 3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
43simplbi2com 1374 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
5 df-mo 1947 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
64, 5sylibr 132 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390  wex 1422  ∃!weu 1943  ∃*wmo 1944
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  mo2icl  2782  rmo2ilem  2914  dffun5r  4981  frecuzrdgtcl  9708  frecuzrdgfunlem  9715
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