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Theorem csbov12g 6838
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov12g (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
Distinct variable group:   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbov12g
StepHypRef Expression
1 csbov123 6836 . 2 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
2 csbconstg 3695 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐹)
32oveqd 6813 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
41, 3syl5eq 2817 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  csb 3682  (class class class)co 6796
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 4924  ax-pow 4975
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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-dm 5260  df-iota 5993  df-fv 6038  df-ov 6799
This theorem is referenced by:  csbov1g  6839  csbov2g  6840  offval22  7408  prmgaplem7  15968  mptscmfsupp0  19138  pm2mp  20850  chfacfscmulfsupp  20884  chfacfpmmulfsupp  20888  cayhamlem4  20913  iccelpart  41894  ply1mulgsumlem4  42702
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