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Theorem csbiotag 4923
Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
csbiotag  |-  ( A  e.  V  ->  [_ 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)    V( x, y)

Proof of Theorem csbiotag
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2883 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota y ph )  = 
[_ A  /  x ]_ ( iota y ph ) )
2 dfsbcq2 2790 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32iotabidv 4916 . . 3  |-  ( z  =  A  ->  ( iota y [ z  /  x ] ph )  =  ( iota y [. A  /  x ]. ph )
)
41, 3eqeq12d 2070 . 2  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )  <->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph ) ) )
5 vex 2577 . . 3  |-  z  e. 
_V
6 nfs1v 1831 . . . 4  |-  F/ x [ z  /  x ] ph
76nfiotaxy 4899 . . 3  |-  F/_ x
( iota y [ z  /  x ] ph )
8 sbequ12 1670 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
98iotabidv 4916 . . 3  |-  ( x  =  z  ->  ( iota y ph )  =  ( iota y [ z  /  x ] ph ) )
105, 7, 9csbief 2919 . 2  |-  [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )
114, 10vtoclg 2630 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota y ph )  =  ( iota y [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   [wsb 1661   [.wsbc 2787   [_csb 2880   iotacio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-sn 3409  df-uni 3609  df-iota 4895
This theorem is referenced by:  csbfv12g  5237
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