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Theorem sb8iota 5570
Description: Variable substitution in description binder. Compare sb8eu 2300. (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 1752 . . . . . 6  |-  F/ w
( ph  <->  x  =  z
)
21sb8 2219 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 2196 . . . . . . 7  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8iota.1 . . . . . . . . 9  |-  F/ y
ph
54nfsb 2236 . . . . . . . 8  |-  F/ y [ w  /  x ] ph
6 equsb3 2228 . . . . . . . . 9  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1752 . . . . . . . . 9  |-  F/ y  w  =  z
86, 7nfxfr 1693 . . . . . . . 8  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1991 . . . . . . 7  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1693 . . . . . 6  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1752 . . . . . 6  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 2171 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 2076 . . . . 5  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 2228 . . . . . . 7  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 2199 . . . . . 6  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1688 . . . . 5  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 275 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817abbii 2557 . . 3  |-  { z  |  A. x (
ph 
<->  x  =  z ) }  =  { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
1918unieqi 4226 . 2  |-  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  U. { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
20 dfiota2 5564 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
21 dfiota2 5564 . 2  |-  ( iota y [ y  /  x ] ph )  = 
U. { z  | 
A. y ( [ y  /  x ] ph 
<->  y  =  z ) }
2219, 20, 213eqtr4i 2462 1  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wal 1436    = wceq 1438   F/wnf 1664   [wsb 1787   {cab 2408   U.cuni 4217   iotacio 5561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rex 2782  df-sn 3998  df-uni 4218  df-iota 5563
This theorem is referenced by: (None)
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