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Theorem csbiota 5590
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 3398 . . . 4  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota y ph )  = 
[_ A  /  x ]_ ( iota y ph ) )
2 dfsbcq2 3302 . . . . 5  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32iotabidv 5582 . . . 4  |-  ( z  =  A  ->  ( iota y [ z  /  x ] ph )  =  ( iota y [. A  /  x ]. ph )
)
41, 3eqeq12d 2444 . . 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 3084 . . . 4  |-  z  e. 
_V
6 nfs1v 2232 . . . . 5  |-  F/ x [ z  /  x ] ph
76nfiota 5565 . . . 4  |-  F/_ x
( iota y [ z  /  x ] ph )
8 sbequ12 2047 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
98iotabidv 5582 . . . 4  |-  ( x  =  z  ->  ( iota y ph )  =  ( iota y [ z  /  x ] ph ) )
105, 7, 9csbief 3420 . . 3  |-  [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )
114, 10vtoclg 3139 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
)
12 csbprc 3798 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  (/) )
13 sbcex 3309 . . . . . 6  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
1413con3i 140 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ph )
1514nexdv 1771 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. y [. A  /  x ]. ph )
16 euex 2290 . . . . 5  |-  ( E! y [. A  /  x ]. ph  ->  E. y [. A  /  x ]. ph )
1716con3i 140 . . . 4  |-  ( -. 
E. y [. A  /  x ]. ph  ->  -.  E! y [. A  /  x ]. ph )
18 iotanul 5576 . . . 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 2466 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph ) )
2111, 20pm2.61i 167 1  |-  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437   E.wex 1659   [wsb 1786    e. wcel 1868   E!weu 2265   _Vcvv 3081   [.wsbc 3299   [_csb 3395   (/)c0 3761   iotacio 5559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-in 3443  df-ss 3450  df-nul 3762  df-sn 3997  df-uni 4217  df-iota 5561
This theorem is referenced by:  csbfv12  5912
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