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Theorem csbaovg 27713
Description: Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
csbaovg  |-  ( A  e.  D  ->  [_ A  /  x ]_ (( B F C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )

Proof of Theorem csbaovg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3197 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ (( B F C))  =  [_ A  /  x ]_ (( B F C))  )
2 csbeq1 3197 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3197 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3197 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4aoveq123d 27711 . . 3  |-  ( y  =  A  -> (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
61, 5eqeq12d 2401 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  <->  [_ A  /  x ]_ (( B F C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
)
7 vex 2902 . . 3  |-  y  e. 
_V
8 nfcsb1v 3226 . . . 4  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 3226 . . . 4  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 3226 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfaov 27712 . . 3  |-  F/_ x (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))
12 csbeq1a 3202 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 3202 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3202 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14aoveq123d 27711 . . 3  |-  ( x  =  y  -> (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  )
167, 11, 15csbief 3235 . 2  |-  [_ y  /  x ]_ (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))
176, 16vtoclg 2954 1  |-  ( A  e.  D  ->  [_ A  /  x ]_ (( B F C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   [_csb 3194   ((caov 27641
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-3an 938  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-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fv 5402  df-dfat 27642  df-afv 27643  df-aov 27644
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