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Theorem csbriotag 5710
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 2978 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B  ph ) )
2 dfsbcq2 2885 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 5700 . . 3  |-  ( z  =  A  ->  ( iota_ y  e.  B  [
z  /  x ] ph )  =  ( iota_ y  e.  B  [. A  /  x ]. ph )
)
41, 3eqeq12d 2132 . 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 2663 . . 3  |-  z  e. 
_V
6 nfs1v 1892 . . . 4  |-  F/ x [ z  /  x ] ph
7 nfcv 2258 . . . 4  |-  F/_ x B
86, 7nfriota 5707 . . 3  |-  F/_ x
( iota_ y  e.  B  [ z  /  x ] ph )
9 sbequ12 1729 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 5700 . . 3  |-  ( x  =  z  ->  ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph ) )
115, 8, 10csbief 3014 . 2  |-  [_ z  /  x ]_ ( iota_ y  e.  B  ph )  =  ( iota_ y  e.  B  [ z  /  x ] ph )
124, 11vtoclg 2720 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 1316    e. wcel 1465   [wsb 1720   [.wsbc 2882   [_csb 2975   iota_crio 5697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-sn 3503  df-uni 3707  df-iota 5058  df-riota 5698
This theorem is referenced by: (None)
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