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Theorem csbovg 6053
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
csbovg  |-  ( A  e.  D  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
) )

Proof of Theorem csbovg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3199 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( B F C )  = 
[_ A  /  x ]_ ( B F C ) )
2 csbeq1 3199 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3199 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3199 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4oveq123d 6043 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
61, 5eqeq12d 2403 . 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 2904 . . 3  |-  y  e. 
_V
8 nfcsb1v 3228 . . . 4  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 3228 . . . 4  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 3228 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfov 6045 . . 3  |-  F/_ x
( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )
12 csbeq1a 3204 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 3204 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3204 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14oveq123d 6043 . . 3  |-  ( x  =  y  ->  ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
) )
167, 11, 15csbief 3237 . 2  |-  [_ y  /  x ]_ ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
)
176, 16vtoclg 2956 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 3196  (class class class)co 6022
This theorem is referenced by:  csbov12g  6054
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 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-ov 6025
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