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Theorem csbriota 6166
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 3392 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B  ph ) )
2 dfsbcq2 3290 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6156 . . . 4  |-  ( z  =  A  ->  ( iota_ y  e.  B  [
z  /  x ] ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
41, 3eqeq12d 2473 . . 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 3074 . . . 4  |-  z  e. 
_V
6 nfs1v 2149 . . . . 5  |-  F/ x [ z  /  x ] ph
7 nfcv 2613 . . . . 5  |-  F/_ x B
86, 7nfriota 6163 . . . 4  |-  F/_ x
( iota_ y  e.  B  [ z  /  x ] ph )
9 sbequ12 1945 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6156 . . . 4  |-  ( x  =  z  ->  ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) )
115, 8, 10csbief 3414 . . 3  |-  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph )
124, 11vtoclg 3129 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
13 csbprc 3774 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  (/) )
14 df-riota 6154 . . . 4  |-  ( iota_ y  e.  B  [. A  /  x ]. ph )  =  ( iota y
( y  e.  B  /\  [. A  /  x ]. ph ) )
15 euex 2288 . . . . . . 7  |-  ( E! y ( y  e.  B  /\  [. A  /  x ]. ph )  ->  E. y ( y  e.  B  /\  [. A  /  x ]. ph )
)
16 sbcex 3297 . . . . . . . . 9  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1716adantl 466 . . . . . . . 8  |-  ( ( y  e.  B  /\  [. A  /  x ]. ph )  ->  A  e.  _V )
1817exlimiv 1689 . . . . . . 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 5497 . . . . 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 2505 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
2413, 23eqtrd 2492 . 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 1370   E.wex 1587   [wsb 1702    e. wcel 1758   E!weu 2260   _Vcvv 3071   [.wsbc 3287   [_csb 3389   (/)c0 3738   iotacio 5480   iota_crio 6153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-sn 3979  df-uni 4193  df-iota 5482  df-riota 6154
This theorem is referenced by:  csbriotagOLD  6167  cdlemkid3N  34886  cdlemkid4  34887
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