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Theorem csbabg 2989
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
csbabg  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { 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 csbabg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbccom 2914 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
2 df-clab 2075 . . . . 5  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
3 sbsbc 2844 . . . . 5  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
42, 3bitri 182 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
5 df-clab 2075 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 2844 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 182 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 2898 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
91, 4, 83bitr4i 210 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2g 2952 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  { y  |  ph }  <->  z  e.  [_ A  /  x ]_ { y  |  ph } ) )
119, 10syl5rbb 191 . 2  |-  ( A  e.  V  ->  (
z  e.  [_ A  /  x ]_ { y  |  ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
) )
1211eqrdv 2086 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   [wsb 1692   {cab 2074   [.wsbc 2840   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  csbsng  3503  csbunig  3661  csbxpg  4519  csbdmg  4630  csbrng  4892
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