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Theorem modc 1986
Description: Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
Hypothesis
Ref Expression
modc.1  |-  F/ y
ph
Assertion
Ref Expression
modc  |-  (DECID  E. x ph  ->  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem modc
StepHypRef Expression
1 modc.1 . . 3  |-  F/ y
ph
21mo23 1984 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
3 exmiddc 778 . . 3  |-  (DECID  E. x ph  ->  ( E. x ph  \/  -.  E. x ph ) )
41mor 1985 . . . 4  |-  ( E. x ph  ->  ( A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  E. y A. x ( ph  ->  x  =  y ) ) )
51mo2n 1971 . . . . 5  |-  ( -. 
E. x ph  ->  E. y A. x (
ph  ->  x  =  y ) )
65a1d 22 . . . 4  |-  ( -. 
E. x ph  ->  ( A. x A. y
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  E. y A. x ( ph  ->  x  =  y ) ) )
74, 6jaoi 669 . . 3  |-  ( ( E. x ph  \/  -.  E. x ph )  ->  ( A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
83, 7syl 14 . 2  |-  (DECID  E. x ph  ->  ( A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  ->  E. y A. x (
ph  ->  x  =  y ) ) )
92, 8impbid2 141 1  |-  (DECID  E. x ph  ->  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776   A.wal 1283   F/wnf 1390   E.wex 1422   [wsb 1687
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688
This theorem is referenced by:  mo2dc  1998
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