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Theorem cbvriota 5478
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 eleq1 2100 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 sbequ12 1654 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
31, 2anbi12d 442 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) ) )
4 nfv 1421 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
5 nfv 1421 . . . . 5  |-  F/ x  z  e.  A
6 nfs1v 1815 . . . . 5  |-  F/ x [ z  /  x ] ph
75, 6nfan 1457 . . . 4  |-  F/ x
( z  e.  A  /\  [ z  /  x ] ph )
83, 4, 7cbviota 4872 . . 3  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )
9 eleq1 2100 . . . . 5  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
10 sbequ 1721 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
11 cbvriota.2 . . . . . . 7  |-  F/ x ps
12 cbvriota.3 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1311, 12sbie 1674 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
1410, 13syl6bb 185 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
159, 14anbi12d 442 . . . 4  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
16 nfv 1421 . . . . 5  |-  F/ y  z  e.  A
17 cbvriota.1 . . . . . 6  |-  F/ y
ph
1817nfsb 1822 . . . . 5  |-  F/ y [ z  /  x ] ph
1916, 18nfan 1457 . . . 4  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
20 nfv 1421 . . . 4  |-  F/ z ( y  e.  A  /\  ps )
2115, 19, 20cbviota 4872 . . 3  |-  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )  =  ( iota y ( y  e.  A  /\  ps ) )
228, 21eqtri 2060 . 2  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota y ( y  e.  A  /\  ps )
)
23 df-riota 5468 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
24 df-riota 5468 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2522, 23, 243eqtr4i 2070 1  |-  ( iota_ x  e.  A  ph )  =  ( iota_ y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   F/wnf 1349    e. wcel 1393   [wsb 1645   iotacio 4865   iota_crio 5467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-sn 3381  df-uni 3581  df-iota 4867  df-riota 5468
This theorem is referenced by:  cbvriotav  5479
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