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Theorem csbaovg 28040
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 3084 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ (( B F C))  =  [_ A  /  x ]_ (( B F C))  )
2 csbeq1 3084 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3084 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3084 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4aoveq123d 28038 . . 3  |-  ( y  =  A  -> (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
61, 5eqeq12d 2297 . 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 2791 . . 3  |-  y  e. 
_V
8 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfaov 28039 . . 3  |-  F/_ x (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))
12 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14aoveq123d 28038 . . 3  |-  ( x  =  y  -> (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  )
167, 11, 15csbief 3122 . 2  |-  [_ y  /  x ]_ (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))
176, 16vtoclg 2843 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 1623    e. wcel 1684   [_csb 3081   ((caov 27973
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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