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Theorem csbov2g 6836
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 6834 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
2 csbconstg 3695 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
32oveq1d 6808 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
41, 3eqtrd 2805 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  csb 3682  (class class class)co 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-dm 5259  df-iota 5994  df-fv 6039  df-ov 6796
This theorem is referenced by:  csbnegg  10480  prmgaplem7  15968  matgsum  20460  scmatscm  20537  pm2mpf1lem  20819  pm2mpcoe1  20825  pm2mpmhmlem2  20844  monmat2matmon  20849  divcncf  23435  logbmpt  24747  finxpreclem4  33568  cotrclrcl  38560  ply1mulgsumlem3  42704  ply1mulgsumlem4  42705  ply1mulgsum  42706
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