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Theorem cbviota 5487
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 1674 . . . . . 6  |-  F/ z ( ph  <->  x  =  w )
2 nfs1v 2149 . . . . . . 7  |-  F/ x [ z  /  x ] ph
3 nfv 1674 . . . . . . 7  |-  F/ x  z  =  w
42, 3nfbi 1869 . . . . . 6  |-  F/ x
( [ z  /  x ] ph  <->  z  =  w )
5 sbequ12 1945 . . . . . . 7  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
6 equequ1 1738 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
75, 6bibi12d 321 . . . . . 6  |-  ( x  =  z  ->  (
( ph  <->  x  =  w
)  <->  ( [ z  /  x ] ph  <->  z  =  w ) ) )
81, 4, 7cbval 1978 . . . . 5  |-  ( A. x ( ph  <->  x  =  w )  <->  A. z
( [ z  /  x ] ph  <->  z  =  w ) )
9 cbviota.2 . . . . . . . 8  |-  F/ y
ph
109nfsb 2153 . . . . . . 7  |-  F/ y [ z  /  x ] ph
11 nfv 1674 . . . . . . 7  |-  F/ y  z  =  w
1210, 11nfbi 1869 . . . . . 6  |-  F/ y ( [ z  /  x ] ph  <->  z  =  w )
13 nfv 1674 . . . . . 6  |-  F/ z ( ps  <->  y  =  w )
14 sbequ 2074 . . . . . . . 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 2107 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<->  ps )
1814, 17syl6bb 261 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
19 equequ1 1738 . . . . . . 7  |-  ( z  =  y  ->  (
z  =  w  <->  y  =  w ) )
2018, 19bibi12d 321 . . . . . 6  |-  ( z  =  y  ->  (
( [ z  /  x ] ph  <->  z  =  w )  <->  ( ps  <->  y  =  w ) ) )
2112, 13, 20cbval 1978 . . . . 5  |-  ( A. z ( [ z  /  x ] ph  <->  z  =  w )  <->  A. y
( ps  <->  y  =  w ) )
228, 21bitri 249 . . . 4  |-  ( A. x ( ph  <->  x  =  w )  <->  A. y
( ps  <->  y  =  w ) )
2322abbii 2585 . . 3  |-  { w  |  A. x ( ph  <->  x  =  w ) }  =  { w  | 
A. y ( ps  <->  y  =  w ) }
2423unieqi 4201 . 2  |-  U. {
w  |  A. x
( ph  <->  x  =  w
) }  =  U. { w  |  A. y ( ps  <->  y  =  w ) }
25 dfiota2 5483 . 2  |-  ( iota
x ph )  =  U. { w  |  A. x ( ph  <->  x  =  w ) }
26 dfiota2 5483 . 2  |-  ( iota y ps )  = 
U. { w  | 
A. y ( ps  <->  y  =  w ) }
2724, 25, 263eqtr4i 2490 1  |-  ( iota
x ph )  =  ( iota y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370   F/wnf 1590   [wsb 1702   {cab 2436   U.cuni 4192   iotacio 5480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-sn 3979  df-uni 4193  df-iota 5482
This theorem is referenced by:  cbviotav  5488  fvopab5  5897  cbvriota  6164
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