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Theorem csbcog 36312
Description: Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
Assertion
Ref Expression
csbcog  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  o.  C )  =  ( [_ A  /  x ]_ B  o.  [_ A  /  x ]_ C
) )

Proof of Theorem csbcog
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( B  o.  C )  = 
[_ A  /  x ]_ ( B  o.  C
) )
2 csbeq1 3352 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3352 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
42, 3coeq12d 5004 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B  o.  [_ y  /  x ]_ C )  =  ( [_ A  /  x ]_ B  o.  [_ A  /  x ]_ C ) )
51, 4eqeq12d 2486 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( B  o.  C
)  =  ( [_ y  /  x ]_ B  o.  [_ y  /  x ]_ C )  <->  [_ A  /  x ]_ ( B  o.  C )  =  (
[_ A  /  x ]_ B  o.  [_ A  /  x ]_ C ) ) )
6 vex 3034 . . 3  |-  y  e. 
_V
7 nfcsb1v 3365 . . . 4  |-  F/_ x [_ y  /  x ]_ B
8 nfcsb1v 3365 . . . 4  |-  F/_ x [_ y  /  x ]_ C
97, 8nfco 5005 . . 3  |-  F/_ x
( [_ y  /  x ]_ B  o.  [_ y  /  x ]_ C )
10 csbeq1a 3358 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3358 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1210, 11coeq12d 5004 . . 3  |-  ( x  =  y  ->  ( B  o.  C )  =  ( [_ y  /  x ]_ B  o.  [_ y  /  x ]_ C ) )
136, 9, 12csbief 3374 . 2  |-  [_ y  /  x ]_ ( B  o.  C )  =  ( [_ y  /  x ]_ B  o.  [_ y  /  x ]_ C
)
145, 13vtoclg 3093 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  o.  C )  =  ( [_ A  /  x ]_ B  o.  [_ A  /  x ]_ C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   [_csb 3349    o. ccom 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-co 4848
This theorem is referenced by:  brtrclfv2  36390
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