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Theorem cbvriota 5506
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 2116 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 sbequ12 1670 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
31, 2anbi12d 450 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) ) )
4 nfv 1437 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
5 nfv 1437 . . . . 5  |-  F/ x  z  e.  A
6 nfs1v 1831 . . . . 5  |-  F/ x [ z  /  x ] ph
75, 6nfan 1473 . . . 4  |-  F/ x
( z  e.  A  /\  [ z  /  x ] ph )
83, 4, 7cbviota 4900 . . 3  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )
9 eleq1 2116 . . . . 5  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
10 sbequ 1737 . . . . . 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 1690 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
1410, 13syl6bb 189 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
159, 14anbi12d 450 . . . 4  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
16 nfv 1437 . . . . 5  |-  F/ y  z  e.  A
17 cbvriota.1 . . . . . 6  |-  F/ y
ph
1817nfsb 1838 . . . . 5  |-  F/ y [ z  /  x ] ph
1916, 18nfan 1473 . . . 4  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
20 nfv 1437 . . . 4  |-  F/ z ( y  e.  A  /\  ps )
2115, 19, 20cbviota 4900 . . 3  |-  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )  =  ( iota y ( y  e.  A  /\  ps ) )
228, 21eqtri 2076 . 2  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota y ( y  e.  A  /\  ps )
)
23 df-riota 5496 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
24 df-riota 5496 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2522, 23, 243eqtr4i 2086 1  |-  ( iota_ x  e.  A  ph )  =  ( iota_ y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365    e. wcel 1409   [wsb 1661   iotacio 4893   iota_crio 5495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-sn 3409  df-uni 3609  df-iota 4895  df-riota 5496
This theorem is referenced by:  cbvriotav  5507
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