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Theorem csbdmg 27980
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 3142 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B } )
2 sbcexg 3041 . . . . 5  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w [. A  /  x ]. <. y ,  w >.  e.  B ) )
3 sbcel2g 3102 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. <. y ,  w >.  e.  B  <->  <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
43exbidv 1612 . . . . 5  |-  ( A  e.  V  ->  ( E. w [. A  /  x ]. <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B ) )
52, 4bitrd 244 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B ) )
65abbidv 2397 . . 3  |-  ( A  e.  V  ->  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
71, 6eqtrd 2315 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
)
8 dfdm3 4867 . . 3  |-  dom  B  =  { y  |  E. w <. y ,  w >.  e.  B }
98csbeq2i 3107 . 2  |-  [_ A  /  x ]_ dom  B  =  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }
10 dfdm3 4867 . 2  |-  dom  [_ A  /  x ]_ B  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4g 2340 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   [.wsbc 2991   [_csb 3081   <.cop 3643   dom cdm 4689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082  df-br 4024  df-dm 4699
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