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Theorem csbiota 5575
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 3366 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota y ph )  = 
[_ A  /  x ]_ ( iota y ph ) )
2 dfsbcq2 3270 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32iotabidv 5567 . . . 4  |-  ( z  =  A  ->  ( iota y [ z  /  x ] ph )  =  ( iota y [. A  /  x ]. ph )
)
41, 3eqeq12d 2466 . . 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 3048 . . . 4  |-  z  e. 
_V
6 nfs1v 2266 . . . . 5  |-  F/ x [ z  /  x ] ph
76nfiota 5550 . . . 4  |-  F/_ x
( iota y [ z  /  x ] ph )
8 sbequ12 2083 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
98iotabidv 5567 . . . 4  |-  ( x  =  z  ->  ( iota y ph )  =  ( iota y [ z  /  x ] ph ) )
105, 7, 9csbief 3388 . . 3  |-  [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )
114, 10vtoclg 3107 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
)
12 csbprc 3770 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  (/) )
13 sbcex 3277 . . . . . 6  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1413con3i 141 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ph )
1514nexdv 1782 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. y [. A  /  x ]. ph )
16 euex 2323 . . . . 5  |-  ( E! y [. A  /  x ]. ph  ->  E. y [. A  /  x ]. ph )
1716con3i 141 . . . 4  |-  ( -. 
E. y [. A  /  x ]. ph  ->  -.  E! y [. A  /  x ]. ph )
18 iotanul 5561 . . . 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 2488 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph ) )
2111, 20pm2.61i 168 1  |-  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1444   E.wex 1663   [wsb 1797    e. wcel 1887   E!weu 2299   _Vcvv 3045   [.wsbc 3267   [_csb 3363   (/)c0 3731   iotacio 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-uni 4199  df-iota 5546
This theorem is referenced by:  csbfv12  5900
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