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Theorem csbdmg 27650
Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
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 3253 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B } )
2 sbcexg 3154 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w [. A  /  x ]. <. y ,  w >.  e.  B ) )
3 sbcel2g 3215 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. <. y ,  w >.  e.  B  <->  <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
43exbidv 1633 . . . . 5  |-  ( A  e.  V  ->  ( E. w [. A  /  x ]. <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B ) )
52, 4bitrd 245 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
65abbidv 2501 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
71, 6eqtrd 2419 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
8 dfdm3 4998 . . 3  |-  dom  B  =  { y  |  E. w <. y ,  w >.  e.  B }
98csbeq2i 3220 . 2  |-  [_ A  /  x ]_ dom  B  =  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }
10 dfdm3 4998 . 2  |-  dom  [_ A  /  x ]_ B  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2444 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   [.wsbc 3104   [_csb 3194   <.cop 3760   dom cdm 4818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105  df-csb 3195  df-br 4154  df-dm 4828
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