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Theorem csbov1g 7478
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov1g (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbov1g
StepHypRef Expression
1 csbov12g 7477 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
2 csbconstg 3927 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32oveq2d 7447 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
41, 3eqtrd 2775 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  csb 3908  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  modfsummods  15826  fprodmodd  16030  scmatscm  22535  idpm2idmp  22823  monmat2matmon  22846  pm2mp  22847  chfacfscmulfsupp  22881  cayhamlem4  22910  iuninc  32581  ellimcabssub0  45573  fsummmodsndifre  47299  fsummmodsnunz  47300  ply1mulgsumlem4  48235
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