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Theorem mo23 1989
Description: An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
Hypothesis
Ref Expression
mo23.1  |-  F/ y
ph
Assertion
Ref Expression
mo23  |-  ( 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 mo23
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mo23.1 . . . . 5  |-  F/ y
ph
2 nfv 1466 . . . . 5  |-  F/ y  x  =  z
31, 2nfim 1509 . . . 4  |-  F/ y ( ph  ->  x  =  z )
43nfal 1513 . . 3  |-  F/ y A. x ( ph  ->  x  =  z )
5 nfv 1466 . . 3  |-  F/ z A. x ( ph  ->  x  =  y )
6 equequ2 1646 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
76imbi2d 228 . . . 4  |-  ( z  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ph  ->  x  =  y ) ) )
87albidv 1752 . . 3  |-  ( z  =  y  ->  ( A. x ( ph  ->  x  =  z )  <->  A. x
( ph  ->  x  =  y ) ) )
94, 5, 8cbvex 1686 . 2  |-  ( E. z A. x (
ph  ->  x  =  z )  <->  E. y A. x
( ph  ->  x  =  y ) )
10 nfs1v 1863 . . . . . . . 8  |-  F/ x [ y  /  x ] ph
11 nfv 1466 . . . . . . . 8  |-  F/ x  y  =  z
1210, 11nfim 1509 . . . . . . 7  |-  F/ x
( [ y  /  x ] ph  ->  y  =  z )
13 sbequ2 1699 . . . . . . . 8  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
14 ax-8 1440 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
1513, 14imim12d 73 . . . . . . 7  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  ->  ( [ y  /  x ] ph  ->  y  =  z ) ) )
163, 12, 15cbv3 1677 . . . . . 6  |-  ( A. x ( ph  ->  x  =  z )  ->  A. y ( [ y  /  x ] ph  ->  y  =  z ) )
1716ancli 316 . . . . 5  |-  ( A. x ( ph  ->  x  =  z )  -> 
( A. x (
ph  ->  x  =  z )  /\  A. y
( [ y  /  x ] ph  ->  y  =  z ) ) )
183nfri 1457 . . . . . 6  |-  ( (
ph  ->  x  =  z )  ->  A. y
( ph  ->  x  =  z ) )
1912nfri 1457 . . . . . 6  |-  ( ( [ y  /  x ] ph  ->  y  =  z )  ->  A. x
( [ y  /  x ] ph  ->  y  =  z ) )
2018, 19aaanh 1523 . . . . 5  |-  ( A. x A. y ( (
ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  <->  ( A. x ( ph  ->  x  =  z )  /\  A. y ( [ y  /  x ] ph  ->  y  =  z ) ) )
2117, 20sylibr 132 . . . 4  |-  ( A. x ( ph  ->  x  =  z )  ->  A. x A. y ( ( ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) ) )
22 prth 336 . . . . . 6  |-  ( ( ( ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  ( x  =  z  /\  y  =  z ) ) )
23 equtr2 1644 . . . . . 6  |-  ( ( x  =  z  /\  y  =  z )  ->  x  =  y )
2422, 23syl6 33 . . . . 5  |-  ( ( ( ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
25242alimi 1390 . . . 4  |-  ( A. x A. y ( (
ph  ->  x  =  z )  /\  ( [ y  /  x ] ph  ->  y  =  z ) )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
2621, 25syl 14 . . 3  |-  ( A. x ( ph  ->  x  =  z )  ->  A. x A. y ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y ) )
2726exlimiv 1534 . 2  |-  ( E. z A. x (
ph  ->  x  =  z )  ->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
289, 27sylbir 133 1  |-  ( 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:    -> wi 4    /\ wa 102   A.wal 1287   F/wnf 1394   E.wex 1426   [wsb 1692
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  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
This theorem is referenced by:  modc  1991  eu2  1992  eu3h  1993
  Copyright terms: Public domain W3C validator