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Theorem csbiotag 5228
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 3075 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota y ph )  = 
[_ A  /  x ]_ ( iota y ph ) )
2 dfsbcq2 2980 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32iotabidv 5218 . . 3  |-  ( z  =  A  ->  ( iota y [ z  /  x ] ph )  =  ( iota y [. A  /  x ]. ph )
)
41, 3eqeq12d 2204 . 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 2755 . . 3  |-  z  e. 
_V
6 nfs1v 1951 . . . 4  |-  F/ x [ z  /  x ] ph
76nfiotaw 5200 . . 3  |-  F/_ x
( iota y [ z  /  x ] ph )
8 sbequ12 1782 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
98iotabidv 5218 . . 3  |-  ( x  =  z  ->  ( iota y ph )  =  ( iota y [ z  /  x ] ph ) )
105, 7, 9csbief 3116 . 2  |-  [_ z  /  x ]_ ( iota y ph )  =  ( iota y [ z  /  x ] ph )
114, 10vtoclg 2812 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 1364   [wsb 1773    e. wcel 2160   [.wsbc 2977   [_csb 3072   iotacio 5194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-sn 3613  df-uni 3825  df-iota 5196
This theorem is referenced by:  csbfv12g  5571
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