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Theorem mo2n 2047
Description: There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
Hypothesis
Ref Expression
mon.1  |-  F/ y
ph
Assertion
Ref Expression
mo2n  |-  ( -. 
E. x ph  ->  E. y A. x (
ph  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo2n
StepHypRef Expression
1 mon.1 . . 3  |-  F/ y
ph
21sb8e 1850 . 2  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 alnex 1492 . . 3  |-  ( A. y  -.  [ y  /  x ] ph  <->  -.  E. y [ y  /  x ] ph )
4 nfs1v 1932 . . . . . 6  |-  F/ x [ y  /  x ] ph
54nfn 1651 . . . . 5  |-  F/ x  -.  [ y  /  x ] ph
61nfn 1651 . . . . 5  |-  F/ y  -.  ph
7 sbequ1 1761 . . . . . . 7  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
87equcoms 1701 . . . . . 6  |-  ( y  =  x  ->  ( ph  ->  [ y  /  x ] ph ) )
98con3d 626 . . . . 5  |-  ( y  =  x  ->  ( -.  [ y  /  x ] ph  ->  -.  ph )
)
105, 6, 9cbv3 1735 . . . 4  |-  ( A. y  -.  [ y  /  x ] ph  ->  A. x  -.  ph )
11 pm2.21 612 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  x  =  y ) )
1211alimi 1448 . . . 4  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  x  =  y ) )
13 19.8a 1583 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) )
1410, 12, 133syl 17 . . 3  |-  ( A. y  -.  [ y  /  x ] ph  ->  E. y A. x ( ph  ->  x  =  y ) )
153, 14sylbir 134 . 2  |-  ( -. 
E. y [ y  /  x ] ph  ->  E. y A. x
( ph  ->  x  =  y ) )
162, 15sylnbi 673 1  |-  ( -. 
E. x ph  ->  E. y A. x (
ph  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1346   F/wnf 1453   E.wex 1485   [wsb 1755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756
This theorem is referenced by:  modc  2062
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