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Theorem csbdm 5197
Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
csbdm  |-  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B

Proof of Theorem csbdm
Dummy variables  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 3855 . . 3  |-  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }
2 sbcex2 3385 . . . . 5  |-  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w [. A  /  x ]. <. y ,  w >.  e.  B )
3 sbcel2 3831 . . . . . 6  |-  ( [. A  /  x ]. <. y ,  w >.  e.  B  <->  <.
y ,  w >.  e. 
[_ A  /  x ]_ B )
43exbii 1644 . . . . 5  |-  ( E. w [. A  /  x ]. <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B )
52, 4bitri 249 . . . 4  |-  ( [. A  /  x ]. E. w <. y ,  w >.  e.  B  <->  E. w <. y ,  w >.  e. 
[_ A  /  x ]_ B )
65abbii 2601 . . 3  |-  { y  |  [. A  /  x ]. E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
71, 6eqtri 2496 . 2  |-  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
8 dfdm3 5190 . . 3  |-  dom  B  =  { y  |  E. w <. y ,  w >.  e.  B }
98csbeq2i 3836 . 2  |-  [_ A  /  x ]_ dom  B  =  [_ A  /  x ]_ { y  |  E. w <. y ,  w >.  e.  B }
10 dfdm3 5190 . 2  |-  dom  [_ A  /  x ]_ B  =  { y  |  E. w <. y ,  w >.  e.  [_ A  /  x ]_ B }
117, 9, 103eqtr4i 2506 1  |-  [_ A  /  x ]_ dom  B  =  dom  [_ A  /  x ]_ B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   [.wsbc 3331   [_csb 3435   <.cop 4033   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-br 4448  df-dm 5009
This theorem is referenced by:  sbcfng  5728
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