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Theorem cbvriota 6331
Description: Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvriota.1  |-  F/ y
ph
cbvriota.2  |-  F/ x ps
cbvriota.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvriota  |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvriota
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvriota.1 . . . 4  |-  F/ y
ph
2 cbvriota.2 . . . 4  |-  F/ x ps
3 cbvriota.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvreu 2775 . . 3  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
5 eleq1 2356 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
6 sbequ12 1872 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
75, 6anbi12d 691 . . . . 5  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) ) )
8 nfv 1609 . . . . 5  |-  F/ z ( x  e.  A  /\  ph )
9 nfv 1609 . . . . . 6  |-  F/ x  z  e.  A
10 nfs1v 2058 . . . . . 6  |-  F/ x [ z  /  x ] ph
119, 10nfan 1783 . . . . 5  |-  F/ x
( z  e.  A  /\  [ z  /  x ] ph )
127, 8, 11cbviota 5240 . . . 4  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )
13 eleq1 2356 . . . . . 6  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
14 sbequ 2013 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
152, 3sbie 1991 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
1614, 15syl6bb 252 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
1713, 16anbi12d 691 . . . . 5  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
18 nfv 1609 . . . . . 6  |-  F/ y  z  e.  A
191nfsb 2061 . . . . . 6  |-  F/ y [ z  /  x ] ph
2018, 19nfan 1783 . . . . 5  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
21 nfv 1609 . . . . 5  |-  F/ z ( y  e.  A  /\  ps )
2217, 20, 21cbviota 5240 . . . 4  |-  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )  =  ( iota y ( y  e.  A  /\  ps ) )
2312, 22eqtri 2316 . . 3  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota y ( y  e.  A  /\  ps )
)
24 abid2 2413 . . . . 5  |-  { x  |  x  e.  A }  =  A
25 abid2 2413 . . . . 5  |-  { y  |  y  e.  A }  =  A
2624, 25eqtr4i 2319 . . . 4  |-  { x  |  x  e.  A }  =  { y  |  y  e.  A }
2726fveq2i 5544 . . 3  |-  ( Undef `  { x  |  x  e.  A } )  =  ( Undef `  {
y  |  y  e.  A } )
284, 23, 27ifbieq12i 3599 . 2  |-  if ( E! x  e.  A  ph ,  ( iota x
( x  e.  A  /\  ph ) ) ,  ( Undef `  { x  |  x  e.  A } ) )  =  if ( E! y  e.  A  ps , 
( iota y ( y  e.  A  /\  ps ) ) ,  (
Undef `  { y  |  y  e.  A }
) )
29 df-riota 6320 . 2  |-  ( iota_ x  e.  A ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
30 df-riota 6320 . 2  |-  ( iota_ y  e.  A ps )  =  if ( E! y  e.  A  ps , 
( iota y ( y  e.  A  /\  ps ) ) ,  (
Undef `  { y  |  y  e.  A }
) )
3128, 29, 303eqtr4i 2326 1  |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1534    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282   E!wreu 2558   ifcif 3578   iotacio 5233   ` cfv 5271   Undefcund 6312   iota_crio 6313
This theorem is referenced by:  cbvriotav  6332
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-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
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