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Theorem sbmo 2002
Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
sbmo  |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbmo
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . . . . 6  |-  F/ x  z  =  w
21sblim 1874 . . . . 5  |-  ( [ y  /  x ]
( ( ph  /\  [ w  /  z ]
ph )  ->  z  =  w )  <->  ( [
y  /  x ]
( ph  /\  [ w  /  z ] ph )  ->  z  =  w ) )
3 sban 1872 . . . . . 6  |-  ( [ y  /  x ]
( ph  /\  [ w  /  z ] ph ) 
<->  ( [ y  /  x ] ph  /\  [
y  /  x ] [ w  /  z ] ph ) )
43imbi1i 236 . . . . 5  |-  ( ( [ y  /  x ] ( ph  /\  [ w  /  z ]
ph )  ->  z  =  w )  <->  ( ( [ y  /  x ] ph  /\  [ y  /  x ] [
w  /  z ]
ph )  ->  z  =  w ) )
5 sbcom2 1906 . . . . . . 7  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  [ w  /  z ] [ y  /  x ] ph )
65anbi2i 445 . . . . . 6  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] [
w  /  z ]
ph )  <->  ( [
y  /  x ] ph  /\  [ w  / 
z ] [ y  /  x ] ph ) )
76imbi1i 236 . . . . 5  |-  ( ( ( [ y  /  x ] ph  /\  [
y  /  x ] [ w  /  z ] ph )  ->  z  =  w )  <->  ( ( [ y  /  x ] ph  /\  [ w  /  z ] [
y  /  x ] ph )  ->  z  =  w ) )
82, 4, 73bitri 204 . . . 4  |-  ( [ y  /  x ]
( ( ph  /\  [ w  /  z ]
ph )  ->  z  =  w )  <->  ( ( [ y  /  x ] ph  /\  [ w  /  z ] [
y  /  x ] ph )  ->  z  =  w ) )
98sbalv 1924 . . 3  |-  ( [ y  /  x ] A. w ( ( ph  /\ 
[ w  /  z ] ph )  ->  z  =  w )  <->  A. w
( ( [ y  /  x ] ph  /\ 
[ w  /  z ] [ y  /  x ] ph )  ->  z  =  w ) )
109sbalv 1924 . 2  |-  ( [ y  /  x ] A. z A. w ( ( ph  /\  [
w  /  z ]
ph )  ->  z  =  w )  <->  A. z A. w ( ( [ y  /  x ] ph  /\  [ w  / 
z ] [ y  /  x ] ph )  ->  z  =  w ) )
11 nfv 1462 . . . 4  |-  F/ w ph
1211mo3 1997 . . 3  |-  ( E* z ph  <->  A. z A. w ( ( ph  /\ 
[ w  /  z ] ph )  ->  z  =  w ) )
1312sbbii 1690 . 2  |-  ( [ y  /  x ] E* z ph  <->  [ y  /  x ] A. z A. w ( ( ph  /\ 
[ w  /  z ] ph )  ->  z  =  w ) )
14 nfv 1462 . . 3  |-  F/ w [ y  /  x ] ph
1514mo3 1997 . 2  |-  ( E* z [ y  /  x ] ph  <->  A. z A. w ( ( [ y  /  x ] ph  /\  [ w  / 
z ] [ y  /  x ] ph )  ->  z  =  w ) )
1610, 13, 153bitr4i 210 1  |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   [wsb 1687   E*wmo 1944
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 1688  df-eu 1946  df-mo 1947
This theorem is referenced by: (None)
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