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Theorem csbov2g 6316
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov2g  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C
) )
Distinct variable groups:    x, B    x, F
Allowed substitution hints:    A( x)    C( x)    V( x)

Proof of Theorem csbov2g
StepHypRef Expression
1 csbov12g 6314 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C
) )
2 csbconstg 3386 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32oveq1d 6293 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C )  =  ( B F
[_ A  /  x ]_ C ) )
41, 3eqtrd 2443 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   [_csb 3373  (class class class)co 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525  ax-pow 4572
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-dm 4833  df-iota 5533  df-fv 5577  df-ov 6281
This theorem is referenced by:  csbnegg  9853  matgsum  19231  scmatscm  19307  pm2mpf1lem  19587  pm2mpcoe1  19593  pm2mpmhmlem2  19612  monmat2matmon  19617  logbmpt  23455  cotrclrcl  35721  divcncf  37054  ply1mulgsumlem3  38499  ply1mulgsumlem4  38500  ply1mulgsum  38501
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