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Theorem sb8iota 5181
Description: Variable substitution in description binder. Compare sb8eu 2039. (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 1528 . . . . . 6  |-  F/ w
( ph  <->  x  =  z
)
21sb8 1856 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  <->  A. w [ w  /  x ] ( ph  <->  x  =  z ) )
3 sbbi 1959 . . . . . . 7  |-  ( [ w  /  x ]
( ph  <->  x  =  z
)  <->  ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z ) )
4 sb8iota.1 . . . . . . . . 9  |-  F/ y
ph
54nfsb 1946 . . . . . . . 8  |-  F/ y [ w  /  x ] ph
6 equsb3 1951 . . . . . . . . 9  |-  ( [ w  /  x ]
x  =  z  <->  w  =  z )
7 nfv 1528 . . . . . . . . 9  |-  F/ y  w  =  z
86, 7nfxfr 1474 . . . . . . . 8  |-  F/ y [ w  /  x ] x  =  z
95, 8nfbi 1589 . . . . . . 7  |-  F/ y ( [ w  /  x ] ph  <->  [ w  /  x ] x  =  z )
103, 9nfxfr 1474 . . . . . 6  |-  F/ y [ w  /  x ] ( ph  <->  x  =  z )
11 nfv 1528 . . . . . 6  |-  F/ w [ y  /  x ] ( ph  <->  x  =  z )
12 sbequ 1840 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] ( ph  <->  x  =  z )  <->  [ y  /  x ] ( ph  <->  x  =  z ) ) )
1310, 11, 12cbval 1754 . . . . 5  |-  ( A. w [ w  /  x ] ( ph  <->  x  =  z )  <->  A. y [ y  /  x ] ( ph  <->  x  =  z ) )
14 equsb3 1951 . . . . . . 7  |-  ( [ y  /  x ]
x  =  z  <->  y  =  z )
1514sblbis 1960 . . . . . 6  |-  ( [ y  /  x ]
( ph  <->  x  =  z
)  <->  ( [ y  /  x ] ph  <->  y  =  z ) )
1615albii 1470 . . . . 5  |-  ( A. y [ y  /  x ] ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
172, 13, 163bitri 206 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  <->  A. y
( [ y  /  x ] ph  <->  y  =  z ) )
1817abbii 2293 . . 3  |-  { z  |  A. x (
ph 
<->  x  =  z ) }  =  { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
1918unieqi 3817 . 2  |-  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  U. { z  |  A. y ( [ y  /  x ] ph  <->  y  =  z ) }
20 dfiota2 5175 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
21 dfiota2 5175 . 2  |-  ( iota y [ y  /  x ] ph )  = 
U. { z  | 
A. y ( [ y  /  x ] ph 
<->  y  =  z ) }
2219, 20, 213eqtr4i 2208 1  |-  ( iota
x ph )  =  ( iota y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wal 1351    = wceq 1353   F/wnf 1460   [wsb 1762   {cab 2163   U.cuni 3807   iotacio 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-sn 3597  df-uni 3808  df-iota 5174
This theorem is referenced by: (None)
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