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Theorem cbvriota 6562
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 2932 . . 3  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
5 eleq1 2498 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
6 sbequ12 1945 . . . . . 6  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
75, 6anbi12d 693 . . . . 5  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) ) )
8 nfv 1630 . . . . 5  |-  F/ z ( x  e.  A  /\  ph )
9 nfv 1630 . . . . . 6  |-  F/ x  z  e.  A
10 nfs1v 2184 . . . . . 6  |-  F/ x [ z  /  x ] ph
119, 10nfan 1847 . . . . 5  |-  F/ x
( z  e.  A  /\  [ z  /  x ] ph )
127, 8, 11cbviota 5425 . . . 4  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )
13 eleq1 2498 . . . . . 6  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
14 sbequ 2113 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
152, 3sbie 2151 . . . . . . 7  |-  ( [ y  /  x ] ph 
<->  ps )
1614, 15syl6bb 254 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
1713, 16anbi12d 693 . . . . 5  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
18 nfv 1630 . . . . . 6  |-  F/ y  z  e.  A
191nfsb 2187 . . . . . 6  |-  F/ y [ z  /  x ] ph
2018, 19nfan 1847 . . . . 5  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
21 nfv 1630 . . . . 5  |-  F/ z ( y  e.  A  /\  ps )
2217, 20, 21cbviota 5425 . . . 4  |-  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )  =  ( iota y ( y  e.  A  /\  ps ) )
2312, 22eqtri 2458 . . 3  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota y ( y  e.  A  /\  ps )
)
24 abid2 2555 . . . . 5  |-  { x  |  x  e.  A }  =  A
25 abid2 2555 . . . . 5  |-  { y  |  y  e.  A }  =  A
2624, 25eqtr4i 2461 . . . 4  |-  { x  |  x  e.  A }  =  { y  |  y  e.  A }
2726fveq2i 5733 . . 3  |-  ( Undef `  { x  |  x  e.  A } )  =  ( Undef `  {
y  |  y  e.  A } )
284, 23, 27ifbieq12i 3762 . 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 6551 . 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 6551 . 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 2468 1  |-  ( iota_ x  e.  A ph )  =  ( iota_ y  e.  A ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   F/wnf 1554    = wceq 1653   [wsb 1659    e. wcel 1726   {cab 2424   E!wreu 2709   ifcif 3741   iotacio 5418   ` cfv 5456   Undefcund 6543   iota_crio 6544
This theorem is referenced by:  cbvriotav  6563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-riota 6551
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