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Theorem mo2r 2000
Description: A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
Hypothesis
Ref Expression
mo2r.1  |-  F/ y
ph
Assertion
Ref Expression
mo2r  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  E* x ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo2r
StepHypRef Expression
1 mo2r.1 . . . . 5  |-  F/ y
ph
21nfri 1457 . . . 4  |-  ( ph  ->  A. y ph )
32eu3h 1993 . . 3  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
43simplbi2com 1378 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ph  ->  E! x ph ) )
5 df-mo 1952 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
64, 5sylibr 132 1  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  E* x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287   F/wnf 1394   E.wex 1426   E!weu 1948   E*wmo 1949
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  mo2icl  2792  rmo2ilem  2926  dffun5r  5014  frecuzrdgtcl  9784  frecuzrdgfunlem  9791
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