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Theorem csbfv2g 6761
Description: Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbfv2g (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
Distinct variable group:   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbfv2g
StepHypRef Expression
1 csbfv12 6760 . 2 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
2 csbconstg 3830 . . 3 (𝐴𝐶𝐴 / 𝑥𝐹 = 𝐹)
32fveq1d 6719 . 2 (𝐴𝐶 → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐹𝐴 / 𝑥𝐵))
41, 3eqtrid 2789 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  csb 3811  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-dm 5561  df-iota 6338  df-fv 6388
This theorem is referenced by:  csbfv  6762  ixpsnval  8581  swrdspsleq  14230  sumeq2ii  15257  fsumabs  15365  prodeq2ii  15475  fprodabs  15536  ixpsnbasval  20247  coe1fzgsumdlem  21222  evl1gsumdlem  21272  pm2mp  21722  cayhamlem4  21785  iuninc  30619  cdlemk39s  38690
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