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Theorem csbabg 3031
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 2956 . . . 4  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
2 df-clab 2104 . . . . 5  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [ z  /  y ] [. A  /  x ]. ph )
3 sbsbc 2886 . . . . 5  |-  ( [ z  /  y ]
[. A  /  x ]. ph  <->  [. z  /  y ]. [. A  /  x ]. ph )
42, 3bitri 183 . . . 4  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. z  / 
y ]. [. A  /  x ]. ph )
5 df-clab 2104 . . . . . 6  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
6 sbsbc 2886 . . . . . 6  |-  ( [ z  /  y ]
ph 
<-> 
[. z  /  y ]. ph )
75, 6bitri 183 . . . . 5  |-  ( z  e.  { y  | 
ph }  <->  [. z  / 
y ]. ph )
87sbcbii 2940 . . . 4  |-  ( [. A  /  x ]. z  e.  { y  |  ph } 
<-> 
[. A  /  x ]. [. z  /  y ]. ph )
91, 4, 83bitr4i 211 . . 3  |-  ( z  e.  { y  | 
[. A  /  x ]. ph }  <->  [. A  /  x ]. z  e.  {
y  |  ph }
)
10 sbcel2g 2994 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  { y  |  ph }  <->  z  e.  [_ A  /  x ]_ { y  |  ph } ) )
119, 10syl5rbb 192 . 2  |-  ( A  e.  V  ->  (
z  e.  [_ A  /  x ]_ { y  |  ph }  <->  z  e.  { y  |  [. A  /  x ]. ph }
) )
1211eqrdv 2115 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   [wsb 1720   {cab 2103   [.wsbc 2882   [_csb 2975
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-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  csbsng  3554  csbunig  3714  csbxpg  4590  csbdmg  4703  csbrng  4970
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