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Theorem csbriota 6270
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 3433 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B  ph ) )
2 dfsbcq2 3330 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6260 . . . 4  |-  ( z  =  A  ->  ( iota_ y  e.  B  [
z  /  x ] ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
41, 3eqeq12d 2479 . . 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 3112 . . . 4  |-  z  e. 
_V
6 nfs1v 2182 . . . . 5  |-  F/ x [ z  /  x ] ph
7 nfcv 2619 . . . . 5  |-  F/_ x B
86, 7nfriota 6267 . . . 4  |-  F/_ x
( iota_ y  e.  B  [ z  /  x ] ph )
9 sbequ12 1993 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6260 . . . 4  |-  ( x  =  z  ->  ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) )
115, 8, 10csbief 3455 . . 3  |-  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph )
124, 11vtoclg 3167 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
13 csbprc 3830 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  (/) )
14 df-riota 6258 . . . 4  |-  ( iota_ y  e.  B  [. A  /  x ]. ph )  =  ( iota y
( y  e.  B  /\  [. A  /  x ]. ph ) )
15 euex 2309 . . . . . . 7  |-  ( E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  E. y ( y  e.  B  /\  [. A  /  x ]. ph )
)
16 sbcex 3337 . . . . . . . . 9  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1716adantl 466 . . . . . . . 8  |-  ( ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
1817exlimiv 1723 . . . . . . 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 5572 . . . . 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 2511 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
2413, 23eqtrd 2498 . 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 1395   E.wex 1613   [wsb 1740    e. wcel 1819   E!weu 2283   _Vcvv 3109   [.wsbc 3327   [_csb 3430   (/)c0 3793   iotacio 5555   iota_crio 6257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-sn 4033  df-uni 4252  df-iota 5557  df-riota 6258
This theorem is referenced by:  cdlemkid3N  36760  cdlemkid4  36761
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