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Theorem cbviota 5356
Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
Hypotheses
Ref Expression
cbviota.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
cbviota.2  |-  F/ y
ph
cbviota.3  |-  F/ x ps
Assertion
Ref Expression
cbviota  |-  ( iota
x ph )  =  ( iota y ps )

Proof of Theorem cbviota
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1626 . . . . . 6  |-  F/ z ( ph  <->  x  =  w )
2 nfs1v 2132 . . . . . . 7  |-  F/ x [ z  /  x ] ph
3 nfv 1626 . . . . . . 7  |-  F/ x  z  =  w
42, 3nfbi 1846 . . . . . 6  |-  F/ x
( [ z  /  x ] ph  <->  z  =  w )
5 sbequ12 1933 . . . . . . 7  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
6 equequ1 1691 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
75, 6bibi12d 313 . . . . . 6  |-  ( x  =  z  ->  (
( ph  <->  x  =  w
)  <->  ( [ z  /  x ] ph  <->  z  =  w ) ) )
81, 4, 7cbval 2010 . . . . 5  |-  ( A. x ( ph  <->  x  =  w )  <->  A. z
( [ z  /  x ] ph  <->  z  =  w ) )
9 cbviota.2 . . . . . . . 8  |-  F/ y
ph
109nfsb 2135 . . . . . . 7  |-  F/ y [ z  /  x ] ph
11 nfv 1626 . . . . . . 7  |-  F/ y  z  =  w
1210, 11nfbi 1846 . . . . . 6  |-  F/ y ( [ z  /  x ] ph  <->  z  =  w )
13 nfv 1626 . . . . . 6  |-  F/ z ( ps  <->  y  =  w )
14 sbequ 2086 . . . . . . . 8  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
15 cbviota.3 . . . . . . . . 9  |-  F/ x ps
16 cbviota.1 . . . . . . . . 9  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1715, 16sbie 2064 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<->  ps )
1814, 17syl6bb 253 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
19 equequ1 1691 . . . . . . 7  |-  ( z  =  y  ->  (
z  =  w  <->  y  =  w ) )
2018, 19bibi12d 313 . . . . . 6  |-  ( z  =  y  ->  (
( [ z  /  x ] ph  <->  z  =  w )  <->  ( ps  <->  y  =  w ) ) )
2112, 13, 20cbval 2010 . . . . 5  |-  ( A. z ( [ z  /  x ] ph  <->  z  =  w )  <->  A. y
( ps  <->  y  =  w ) )
228, 21bitri 241 . . . 4  |-  ( A. x ( ph  <->  x  =  w )  <->  A. y
( ps  <->  y  =  w ) )
2322abbii 2492 . . 3  |-  { w  |  A. x ( ph  <->  x  =  w ) }  =  { w  | 
A. y ( ps  <->  y  =  w ) }
2423unieqi 3960 . 2  |-  U. {
w  |  A. x
( ph  <->  x  =  w
) }  =  U. { w  |  A. y ( ps  <->  y  =  w ) }
25 dfiota2 5352 . 2  |-  ( iota
x ph )  =  U. { w  |  A. x ( ph  <->  x  =  w ) }
26 dfiota2 5352 . 2  |-  ( iota y ps )  = 
U. { w  | 
A. y ( ps  <->  y  =  w ) }
2724, 25, 263eqtr4i 2410 1  |-  ( iota
x ph )  =  ( iota y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   F/wnf 1550    = wceq 1649   [wsb 1655   {cab 2366   U.cuni 3950   iotacio 5349
This theorem is referenced by:  cbviotav  5357  fvopab5  6463  cbvriota  6489
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 2361
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-sn 3756  df-uni 3951  df-iota 5351
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