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Theorem sb8iota 5367
Description: Variable substitution in description binder. Compare sb8eu 2258. (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 1626 . . . . . 6  |-  F/ w
( ph  <->  x  =  z
)
21sb8 2127 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 2106 . . . . . . 7  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8iota.1 . . . . . . . . 9  |-  F/ y
ph
54nfsb 2144 . . . . . . . 8  |-  F/ y [ w  /  x ] ph
6 equsb3 2137 . . . . . . . . 9  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1626 . . . . . . . . 9  |-  F/ y  w  =  z
86, 7nfxfr 1576 . . . . . . . 8  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1846 . . . . . . 7  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1576 . . . . . 6  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1626 . . . . . 6  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 2095 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 2024 . . . . 5  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 2137 . . . . . . 7  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 2107 . . . . . 6  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1572 . . . . 5  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 263 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817abbii 2501 . . 3  |-  { z  |  A. x (
ph 
<->  x  =  z ) }  =  { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
1918unieqi 3969 . 2  |-  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  U. { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
20 dfiota2 5361 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
21 dfiota2 5361 . 2  |-  ( iota y [ y  /  x ] ph )  = 
U. { z  | 
A. y ( [ y  /  x ] ph 
<->  y  =  z ) }
2219, 20, 213eqtr4i 2419 1  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546   F/wnf 1550    = wceq 1649   [wsb 1655   {cab 2375   U.cuni 3959   iotacio 5358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-sn 3765  df-uni 3960  df-iota 5360
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