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Theorem csbov1g 6834
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 6833 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
2 csbconstg 3693 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32oveq2d 6808 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
41, 3eqtrd 2804 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  csb 3680  (class class class)co 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920  ax-pow 4971
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-dm 5259  df-iota 5994  df-fv 6039  df-ov 6795
This theorem is referenced by:  modfsummods  14731  fprodmodd  14933  scmatscm  20536  idpm2idmp  20825  monmat2matmon  20848  pm2mp  20849  chfacfscmulfsupp  20883  cayhamlem4  20912  iuninc  29711  ellimcabssub0  40361  fsummmodsndifre  41862  fsummmodsnunz  41863  ply1mulgsumlem4  42695
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