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Theorem csbriota 6255
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 3438 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B  ph ) )
2 dfsbcq2 3334 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6245 . . . 4  |-  ( z  =  A  ->  ( iota_ y  e.  B  [
z  /  x ] ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
41, 3eqeq12d 2489 . . 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 3116 . . . 4  |-  z  e. 
_V
6 nfs1v 2164 . . . . 5  |-  F/ x [ z  /  x ] ph
7 nfcv 2629 . . . . 5  |-  F/_ x B
86, 7nfriota 6252 . . . 4  |-  F/_ x
( iota_ y  e.  B  [ z  /  x ] ph )
9 sbequ12 1961 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6245 . . . 4  |-  ( x  =  z  ->  ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) )
115, 8, 10csbief 3460 . . 3  |-  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph )
124, 11vtoclg 3171 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
13 csbprc 3821 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  (/) )
14 df-riota 6243 . . . 4  |-  ( iota_ y  e.  B  [. A  /  x ]. ph )  =  ( iota y
( y  e.  B  /\  [. A  /  x ]. ph ) )
15 euex 2303 . . . . . . 7  |-  ( E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  E. y ( y  e.  B  /\  [. A  /  x ]. ph )
)
16 sbcex 3341 . . . . . . . . 9  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1716adantl 466 . . . . . . . 8  |-  ( ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
1817exlimiv 1698 . . . . . . 7  |-  ( E. y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
1915, 18syl 16 . . . . . 6  |-  ( E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
2019con3i 135 . . . . 5  |-  ( -.  A  e.  _V  ->  -.  E! y ( y  e.  B  /\  [. A  /  x ]. ph )
)
21 iotanul 5564 . . . . 5  |-  ( -.  E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  ( iota y ( y  e.  B  /\  [. A  /  x ]. ph ) )  =  (/) )
2220, 21syl 16 . . . 4  |-  ( -.  A  e.  _V  ->  ( iota y ( y  e.  B  /\  [. A  /  x ]. ph )
)  =  (/) )
2314, 22syl5req 2521 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
2413, 23eqtrd 2508 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
2512, 24pm2.61i 164 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 369    = wceq 1379   E.wex 1596   [wsb 1711    e. wcel 1767   E!weu 2275   _Vcvv 3113   [.wsbc 3331   [_csb 3435   (/)c0 3785   iotacio 5547   iota_crio 6242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-uni 4246  df-iota 5549  df-riota 6243
This theorem is referenced by:  csbriotagOLD  6256  cdlemkid3N  35729  cdlemkid4  35730
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