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Theorem mo2r 1949
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 1409 . . . 4 (φyφ)
32eu3h 1942 . . 3 (∃!xφ ↔ (xφ yx(φx = y)))
43simplbi2com 1330 . 2 (yx(φx = y) → (xφ∃!xφ))
5 df-mo 1901 . 2 (∃*xφ ↔ (xφ∃!xφ))
64, 5sylibr 137 1 (yx(φx = y) → ∃*xφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  wnf 1346  wex 1378  ∃!weu 1897  ∃*wmo 1898
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by:  mo2icl  2714  rmo2ilem  2841  dffun5r  4857  frecuzrdgfn  8859
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