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Theorem sb8iota 4902
Description: Variable substitution in description binder. Compare sb8eu 1929. (Contributed by NM, 18-Mar-2013.)
Hypothesis
Ref Expression
sb8iota.1  |-  F/ y
ph
Assertion
Ref Expression
sb8iota  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )

Proof of Theorem sb8iota
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . . . 6  |-  F/ w
( ph  <->  x  =  z
)
21sb8 1752 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 1849 . . . . . . 7  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8iota.1 . . . . . . . . 9  |-  F/ y
ph
54nfsb 1838 . . . . . . . 8  |-  F/ y [ w  /  x ] ph
6 equsb3 1841 . . . . . . . . 9  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1437 . . . . . . . . 9  |-  F/ y  w  =  z
86, 7nfxfr 1379 . . . . . . . 8  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1497 . . . . . . 7  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1379 . . . . . 6  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1437 . . . . . 6  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 1737 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 1653 . . . . 5  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 1841 . . . . . . 7  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 1850 . . . . . 6  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1375 . . . . 5  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 199 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817abbii 2169 . . 3  |-  { z  |  A. x (
ph 
<->  x  =  z ) }  =  { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
1918unieqi 3618 . 2  |-  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  U. { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
20 dfiota2 4896 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
21 dfiota2 4896 . 2  |-  ( iota y [ y  /  x ] ph )  = 
U. { z  | 
A. y ( [ y  /  x ] ph 
<->  y  =  z ) }
2219, 20, 213eqtr4i 2086 1  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 102   A.wal 1257    = wceq 1259   F/wnf 1365   [wsb 1661   {cab 2042   U.cuni 3608   iotacio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-sn 3409  df-uni 3609  df-iota 4895
This theorem is referenced by: (None)
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