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Theorem csbov12g 6329
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov12g  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C
) )
Distinct variable group:    x, F
Allowed substitution hints:    A( x)    B( x)    C( x)    V( x)

Proof of Theorem csbov12g
StepHypRef Expression
1 csbov123 6326 . 2  |-  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
)
2 csbconstg 3453 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ F  =  F )
32oveqd 6312 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( [_ A  /  x ]_ B F
[_ A  /  x ]_ C ) )
41, 3syl5eq 2520 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   [_csb 3440  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582  ax-pow 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602  df-ov 6298
This theorem is referenced by:  csbov1g  6330  csbov2g  6331  offval22  6874  mptscmfsupp0  17447  pm2mp  19195  chfacfscmulfsupp  19229  chfacfpmmulfsupp  19233  cayhamlem4  19258  ply1mulgsumlem4  32471
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