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Theorem csbov2g 7201
 Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov2g (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbov2g
StepHypRef Expression
1 csbov12g 7199 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
2 csbconstg 3826 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
32oveq1d 7170 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
41, 3eqtrd 2793 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ⦋csb 3807  (class class class)co 7155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-dm 5537  df-iota 6298  df-fv 6347  df-ov 7158 This theorem is referenced by:  csbnegg  10926  prmgaplem7  16453  matgsum  21142  scmatscm  21218  pm2mpf1lem  21499  pm2mpcoe1  21505  pm2mpmhmlem2  21524  monmat2matmon  21529  divcncf  24152  logbmpt  25478  finxpreclem4  35117  cotrclrcl  40844  ply1mulgsumlem3  45190  ply1mulgsumlem4  45191  ply1mulgsum  45192
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