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Theorem mo2r 1994
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 1453 . . . 4  |-  ( ph  ->  A. y ph )
32eu3h 1987 . . 3  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
43simplbi2com 1374 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ph  ->  E! x ph ) )
5 df-mo 1946 . 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 1283   F/wnf 1390   E.wex 1422   E!weu 1942   E*wmo 1943
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 1687  df-eu 1945  df-mo 1946
This theorem is referenced by:  mo2icl  2772  rmo2ilem  2904  dffun5r  4944  frecuzrdgtcl  9494  frecuzrdgfunlem  9501
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