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Theorem csbov123g 5574
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
csbov123g  |-  ( A  e.  D  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
) )

Proof of Theorem csbov123g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2912 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( B F C )  = 
[_ A  /  x ]_ ( B F C ) )
2 csbeq1 2912 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 2912 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 2912 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4oveq123d 5564 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
61, 5eqeq12d 2096 . 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 2605 . . 3  |-  y  e. 
_V
8 nfcsb1v 2939 . . . 4  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 2939 . . . 4  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 2939 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfov 5566 . . 3  |-  F/_ x
( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )
12 csbeq1a 2917 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 2917 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 2917 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14oveq123d 5564 . . 3  |-  ( x  =  y  ->  ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
) )
167, 11, 15csbief 2948 . 2  |-  [_ y  /  x ]_ ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
)
176, 16vtoclg 2659 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 1285    e. wcel 1434   [_csb 2909  (class class class)co 5543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  csbov12g  5575
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