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Theorem csbriotag 5480
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem csbriotag
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2855 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B  ph ) )
2 dfsbcq2 2767 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 5470 . . 3  |-  ( z  =  A  ->  ( iota_ y  e.  B  [
z  /  x ] ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
41, 3eqeq12d 2054 . 2  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) 
<-> 
[_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
) )
5 vex 2560 . . 3  |-  z  e. 
_V
6 nfs1v 1815 . . . 4  |-  F/ x [ z  /  x ] ph
7 nfcv 2178 . . . 4  |-  F/_ x B
86, 7nfriota 5477 . . 3  |-  F/_ x
( iota_ y  e.  B  [ z  /  x ] ph )
9 sbequ12 1654 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 5470 . . 3  |-  ( x  =  z  ->  ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) )
115, 8, 10csbief 2891 . 2  |-  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph )
124, 11vtoclg 2613 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   [wsb 1645   [.wsbc 2764   [_csb 2852   iota_crio 5467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-sn 3381  df-uni 3581  df-iota 4867  df-riota 5468
This theorem is referenced by: (None)
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