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Theorem csbriota 6289
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
Assertion
Ref Expression
csbriota  |-  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem csbriota
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3378 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B  ph ) )
2 dfsbcq2 3282 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6279 . . . 4  |-  ( z  =  A  ->  ( iota_ y  e.  B  [
z  /  x ] ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
41, 3eqeq12d 2477 . . 3  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) 
<-> 
[_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
) )
5 vex 3060 . . . 4  |-  z  e. 
_V
6 nfs1v 2277 . . . . 5  |-  F/ x [ z  /  x ] ph
7 nfcv 2603 . . . . 5  |-  F/_ x B
86, 7nfriota 6286 . . . 4  |-  F/_ x
( iota_ y  e.  B  [ z  /  x ] ph )
9 sbequ12 2094 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6279 . . . 4  |-  ( x  =  z  ->  ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) )
115, 8, 10csbief 3400 . . 3  |-  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph )
124, 11vtoclg 3119 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
13 csbprc 3782 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  (/) )
14 df-riota 6277 . . . 4  |-  ( iota_ y  e.  B  [. A  /  x ]. ph )  =  ( iota y
( y  e.  B  /\  [. A  /  x ]. ph ) )
15 euex 2334 . . . . . . 7  |-  ( E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  E. y ( y  e.  B  /\  [. A  /  x ]. ph )
)
16 sbcex 3289 . . . . . . . . 9  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1716adantl 472 . . . . . . . 8  |-  ( ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
1817exlimiv 1787 . . . . . . 7  |-  ( E. y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
1915, 18syl 17 . . . . . 6  |-  ( E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
2019con3i 142 . . . . 5  |-  ( -.  A  e.  _V  ->  -.  E! y ( y  e.  B  /\  [. A  /  x ]. ph )
)
21 iotanul 5580 . . . . 5  |-  ( -.  E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  ( iota y ( y  e.  B  /\  [. A  /  x ]. ph ) )  =  (/) )
2220, 21syl 17 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota y ( y  e.  B  /\  [. A  /  x ]. ph )
)  =  (/) )
2314, 22syl5req 2509 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
2413, 23eqtrd 2496 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
2512, 24pm2.61i 169 1  |-  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 375    = wceq 1455   E.wex 1674   [wsb 1808    e. wcel 1898   E!weu 2310   _Vcvv 3057   [.wsbc 3279   [_csb 3375   (/)c0 3743   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-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3744  df-sn 3981  df-uni 4213  df-iota 5565  df-riota 6277
This theorem is referenced by:  cdlemkid3N  34545  cdlemkid4  34546
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