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Theorem csbov12g 7402
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 7400 . 2 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)
2 csbconstg 3875 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐹)
32oveqd 7375 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
41, 3eqtrid 2789 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  csb 3856  (class class class)co 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  csbov1g  7403  csbov2g  7404  offval22  8021  prmgaplem7  16930  mptscmfsupp0  20390  pm2mp  22177  chfacfscmulfsupp  22211  chfacfpmmulfsupp  22215  cayhamlem4  22240  iccelpart  45632  ply1mulgsumlem4  46477
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