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Theorem mo2r 1952
 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 1412 . . . 4 (𝜑 → ∀𝑦𝜑)
32eu3h 1945 . . 3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
43simplbi2com 1333 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
5 df-mo 1904 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
64, 5sylibr 137 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  ∃!weu 1900  ∃*wmo 1901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904 This theorem is referenced by:  mo2icl  2720  rmo2ilem  2847  dffun5r  4914  frecuzrdgfn  9198
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