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Theorem cbviota 5240
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 1609 . . . . . 6  |-  F/ z ( ph  <->  x  =  w )
2 nfs1v 2058 . . . . . . 7  |-  F/ x [ z  /  x ] ph
3 nfv 1609 . . . . . . 7  |-  F/ x  z  =  w
42, 3nfbi 1784 . . . . . 6  |-  F/ x
( [ z  /  x ] ph  <->  z  =  w )
5 sbequ12 1872 . . . . . . 7  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
6 equequ1 1667 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
75, 6bibi12d 312 . . . . . 6  |-  ( x  =  z  ->  (
( ph  <->  x  =  w
)  <->  ( [ z  /  x ] ph  <->  z  =  w ) ) )
81, 4, 7cbval 1937 . . . . 5  |-  ( A. x ( ph  <->  x  =  w )  <->  A. z
( [ z  /  x ] ph  <->  z  =  w ) )
9 cbviota.2 . . . . . . . 8  |-  F/ y
ph
109nfsb 2061 . . . . . . 7  |-  F/ y [ z  /  x ] ph
11 nfv 1609 . . . . . . 7  |-  F/ y  z  =  w
1210, 11nfbi 1784 . . . . . 6  |-  F/ y ( [ z  /  x ] ph  <->  z  =  w )
13 nfv 1609 . . . . . 6  |-  F/ z ( ps  <->  y  =  w )
14 sbequ 2013 . . . . . . . 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 1991 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<->  ps )
1814, 17syl6bb 252 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
19 equequ1 1667 . . . . . . 7  |-  ( z  =  y  ->  (
z  =  w  <->  y  =  w ) )
2018, 19bibi12d 312 . . . . . 6  |-  ( z  =  y  ->  (
( [ z  /  x ] ph  <->  z  =  w )  <->  ( ps  <->  y  =  w ) ) )
2112, 13, 20cbval 1937 . . . . 5  |-  ( A. z ( [ z  /  x ] ph  <->  z  =  w )  <->  A. y
( ps  <->  y  =  w ) )
228, 21bitri 240 . . . 4  |-  ( A. x ( ph  <->  x  =  w )  <->  A. y
( ps  <->  y  =  w ) )
2322abbii 2408 . . 3  |-  { w  |  A. x ( ph  <->  x  =  w ) }  =  { w  | 
A. y ( ps  <->  y  =  w ) }
2423unieqi 3853 . 2  |-  U. {
w  |  A. x
( ph  <->  x  =  w
) }  =  U. { w  |  A. y ( ps  <->  y  =  w ) }
25 dfiota2 5236 . 2  |-  ( iota
x ph )  =  U. { w  |  A. x ( ph  <->  x  =  w ) }
26 dfiota2 5236 . 2  |-  ( iota y ps )  = 
U. { w  | 
A. y ( ps  <->  y  =  w ) }
2724, 25, 263eqtr4i 2326 1  |-  ( iota
x ph )  =  ( iota y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   F/wnf 1534    = wceq 1632   [wsb 1638   {cab 2282   U.cuni 3843   iotacio 5233
This theorem is referenced by:  cbviotav  5241  fvopab5  6305  cbvriota  6331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-sn 3659  df-uni 3844  df-iota 5235
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