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Theorem csbiota 5582
Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbiota  |-  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem csbiota
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota y ph )  = 
[_ A  /  x ]_ ( iota y ph ) )
2 dfsbcq2 3258 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32iotabidv 5574 . . . 4  |-  ( z  =  A  ->  ( iota y [ z  /  x ] ph )  =  ( iota y [. A  /  x ]. ph )
)
41, 3eqeq12d 2486 . . 3  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )  <->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph ) ) )
5 vex 3034 . . . 4  |-  z  e. 
_V
6 nfs1v 2286 . . . . 5  |-  F/ x [ z  /  x ] ph
76nfiota 5557 . . . 4  |-  F/_ x
( iota y [ z  /  x ] ph )
8 sbequ12 2098 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
98iotabidv 5574 . . . 4  |-  ( x  =  z  ->  ( iota y ph )  =  ( iota y [ z  /  x ] ph ) )
105, 7, 9csbief 3374 . . 3  |-  [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )
114, 10vtoclg 3093 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
)
12 csbprc 3774 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  (/) )
13 sbcex 3265 . . . . . 6  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1413con3i 142 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ph )
1514nexdv 1790 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. y [. A  /  x ]. ph )
16 euex 2343 . . . . 5  |-  ( E! y [. A  /  x ]. ph  ->  E. y [. A  /  x ]. ph )
1716con3i 142 . . . 4  |-  ( -. 
E. y [. A  /  x ]. ph  ->  -.  E! y [. A  /  x ]. ph )
18 iotanul 5568 . . . 4  |-  ( -.  E! y [. A  /  x ]. ph  ->  ( iota y [. A  /  x ]. ph )  =  (/) )
1915, 17, 183syl 18 . . 3  |-  ( -.  A  e.  _V  ->  ( iota y [. A  /  x ]. ph )  =  (/) )
2012, 19eqtr4d 2508 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph ) )
2111, 20pm2.61i 169 1  |-  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452   E.wex 1671   [wsb 1805    e. wcel 1904   E!weu 2319   _Vcvv 3031   [.wsbc 3255   [_csb 3349   (/)c0 3722   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960  df-uni 4191  df-iota 5553
This theorem is referenced by:  csbfv12  5914
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