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Theorem csbiota 5519
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 3399 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota y ph )  = 
[_ A  /  x ]_ ( iota y ph ) )
2 dfsbcq2 3297 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32iotabidv 5511 . . . 4  |-  ( z  =  A  ->  ( iota y [ z  /  x ] ph )  =  ( iota y [. A  /  x ]. ph )
)
41, 3eqeq12d 2476 . . 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 3081 . . . 4  |-  z  e. 
_V
6 nfs1v 2151 . . . . 5  |-  F/ x [ z  /  x ] ph
76nfiota 5494 . . . 4  |-  F/_ x
( iota y [ z  /  x ] ph )
8 sbequ12 1948 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
98iotabidv 5511 . . . 4  |-  ( x  =  z  ->  ( iota y ph )  =  ( iota y [ z  /  x ] ph ) )
105, 7, 9csbief 3421 . . 3  |-  [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )
114, 10vtoclg 3136 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
)
12 csbprc 3782 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  (/) )
13 sbcex 3304 . . . . . 6  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1413con3i 135 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ph )
1514nexdv 1823 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. y [. A  /  x ]. ph )
16 euex 2290 . . . . 5  |-  ( E! y [. A  /  x ]. ph  ->  E. y [. A  /  x ]. ph )
1716con3i 135 . . . 4  |-  ( -. 
E. y [. A  /  x ]. ph  ->  -.  E! y [. A  /  x ]. ph )
18 iotanul 5505 . . . 4  |-  ( -.  E! y [. A  /  x ]. ph  ->  ( iota y [. A  /  x ]. ph )  =  (/) )
1915, 17, 183syl 20 . . 3  |-  ( -.  A  e.  _V  ->  ( iota y [. A  /  x ]. ph )  =  (/) )
2012, 19eqtr4d 2498 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph ) )
2111, 20pm2.61i 164 1  |-  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370   E.wex 1587   [wsb 1702    e. wcel 1758   E!weu 2262   _Vcvv 3078   [.wsbc 3294   [_csb 3396   (/)c0 3746   iotacio 5488
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 1955  ax-ext 2432
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 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747  df-sn 3987  df-uni 4201  df-iota 5490
This theorem is referenced by:  csbfv12  5835
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