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Theorem csbdmg 4821
Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
csbdmg  |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )

Proof of Theorem csbdmg
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 3118 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B } )
2 sbcex2 3016 . . . . 5  |-  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w [. A  /  x ]. <. y ,  w >.  e.  B )
3 sbcel2g 3078 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. <. y ,  w >.  e.  B  <->  <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
43exbidv 1825 . . . . 5  |-  ( A  e.  V  ->  ( E. w [. A  /  x ]. <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B ) )
52, 4bitrid 192 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
65abbidv 2295 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
71, 6eqtrd 2210 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
8 dfdm3 4814 . . 3  |-  dom  B  =  { y  |  E. w <. y ,  w >.  e.  B }
98csbeq2i 3084 . 2  |-  [_ A  /  x ]_ dom  B  =  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }
10 dfdm3 4814 . 2  |-  dom  [_ A  /  x ]_ B  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2235 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   [.wsbc 2962   [_csb 3057   <.cop 3595   dom cdm 4626
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058  df-br 4004  df-dm 4636
This theorem is referenced by:  sbcfng  5363
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