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Theorem csbaovg 28011
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 3246 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ (( B F C))  =  [_ A  /  x ]_ (( B F C))  )
2 csbeq1 3246 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3246 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3246 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4aoveq123d 28009 . . 3  |-  ( y  =  A  -> (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
61, 5eqeq12d 2449 . 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 2951 . . 3  |-  y  e. 
_V
8 nfcsb1v 3275 . . . 4  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 3275 . . . 4  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 3275 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfaov 28010 . . 3  |-  F/_ x (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))
12 csbeq1a 3251 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 3251 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3251 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14aoveq123d 28009 . . 3  |-  ( x  =  y  -> (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))  )
167, 11, 15csbief 3284 . 2  |-  [_ y  /  x ]_ (( B F C))  = (( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C))
176, 16vtoclg 3003 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 1652    e. wcel 1725   [_csb 3243   ((caov 27940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-dfat 27941  df-afv 27942  df-aov 27943
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