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Theorem cbvriota 6287
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 2528 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2 sbequ12 2094 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
31, 2anbi12d 722 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  A  /\  [ z  /  x ] ph ) ) )
4 nfv 1772 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
5 nfv 1772 . . . . 5  |-  F/ x  z  e.  A
6 nfs1v 2277 . . . . 5  |-  F/ x [ z  /  x ] ph
75, 6nfan 2022 . . . 4  |-  F/ x
( z  e.  A  /\  [ z  /  x ] ph )
83, 4, 7cbviota 5570 . . 3  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )
9 eleq1 2528 . . . . 5  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
10 sbequ 2216 . . . . . 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 2248 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
1410, 13syl6bb 269 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
159, 14anbi12d 722 . . . 4  |-  ( z  =  y  ->  (
( z  e.  A  /\  [ z  /  x ] ph )  <->  ( y  e.  A  /\  ps )
) )
16 nfv 1772 . . . . 5  |-  F/ y  z  e.  A
17 cbvriota.1 . . . . . 6  |-  F/ y
ph
1817nfsb 2280 . . . . 5  |-  F/ y [ z  /  x ] ph
1916, 18nfan 2022 . . . 4  |-  F/ y ( z  e.  A  /\  [ z  /  x ] ph )
20 nfv 1772 . . . 4  |-  F/ z ( y  e.  A  /\  ps )
2115, 19, 20cbviota 5570 . . 3  |-  ( iota z ( z  e.  A  /\  [ z  /  x ] ph ) )  =  ( iota y ( y  e.  A  /\  ps ) )
228, 21eqtri 2484 . 2  |-  ( iota
x ( x  e.  A  /\  ph )
)  =  ( iota y ( y  e.  A  /\  ps )
)
23 df-riota 6277 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
24 df-riota 6277 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2522, 23, 243eqtr4i 2494 1  |-  ( iota_ x  e.  A  ph )  =  ( iota_ y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455   F/wnf 1678   [wsb 1808    e. wcel 1898   iotacio 5563   iota_crio 6276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rex 2755  df-sn 3981  df-uni 4213  df-iota 5565  df-riota 6277
This theorem is referenced by:  cbvriotav  6288  disjinfi  37506
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