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Theorem csbov2g 6311
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 6309 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C
) )
2 csbconstg 3441 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32oveq1d 6290 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C )  =  ( B F
[_ A  /  x ]_ C ) )
41, 3eqtrd 2501 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 1374    e. wcel 1762   [_csb 3428  (class class class)co 6275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569  ax-pow 4618
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-dm 5002  df-iota 5542  df-fv 5587  df-ov 6278
This theorem is referenced by:  csbnegg  9806  matgsum  18699  scmatscm  18775  pm2mpf1lem  19055  pm2mpcoe1  19061  pm2mpmhmlem2  19080  monmat2matmon  19085  divcncf  31177  ply1mulgsumlem3  31936  ply1mulgsumlem4  31937  ply1mulgsum  31938
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