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Theorem mo23 1984
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 1462 . . . . 5  |-  F/ y  x  =  z
31, 2nfim 1505 . . . 4  |-  F/ y ( ph  ->  x  =  z )
43nfal 1509 . . 3  |-  F/ y A. x ( ph  ->  x  =  z )
5 nfv 1462 . . 3  |-  F/ z A. x ( ph  ->  x  =  y )
6 equequ2 1641 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
76imbi2d 228 . . . 4  |-  ( z  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ph  ->  x  =  y ) ) )
87albidv 1747 . . 3  |-  ( z  =  y  ->  ( A. x ( ph  ->  x  =  z )  <->  A. x
( ph  ->  x  =  y ) ) )
94, 5, 8cbvex 1681 . 2  |-  ( E. z A. x (
ph  ->  x  =  z )  <->  E. y A. x
( ph  ->  x  =  y ) )
10 nfs1v 1858 . . . . . . . 8  |-  F/ x [ y  /  x ] ph
11 nfv 1462 . . . . . . . 8  |-  F/ x  y  =  z
1210, 11nfim 1505 . . . . . . 7  |-  F/ x
( [ y  /  x ] ph  ->  y  =  z )
13 sbequ2 1694 . . . . . . . 8  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
14 ax-8 1436 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
1513, 14imim12d 73 . . . . . . 7  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  ->  ( [ y  /  x ] ph  ->  y  =  z ) ) )
163, 12, 15cbv3 1672 . . . . . 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 1453 . . . . . 6  |-  ( (
ph  ->  x  =  z )  ->  A. y
( ph  ->  x  =  z ) )
1912nfri 1453 . . . . . 6  |-  ( ( [ y  /  x ] ph  ->  y  =  z )  ->  A. x
( [ y  /  x ] ph  ->  y  =  z ) )
2018, 19aaanh 1519 . . . . 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 1639 . . . . . 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 1386 . . . 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 1530 . 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 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-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-nf 1391  df-sb 1688
This theorem is referenced by:  modc  1986  eu2  1987  eu3h  1988
  Copyright terms: Public domain W3C validator