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Theorem csbfv2g 6956
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 6955 . 2 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
2 csbconstg 3927 . . 3 (𝐴𝐶𝐴 / 𝑥𝐹 = 𝐹)
32fveq1d 6909 . 2 (𝐴𝐶 → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐹𝐴 / 𝑥𝐵))
41, 3eqtrid 2787 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  csb 3908  cfv 6563
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
This theorem is referenced by:  csbfv  6957  ixpsnval  8939  swrdspsleq  14700  sumeq2ii  15726  fsumabs  15834  prodeq2ii  15944  fprodabs  16007  ixpsnbasval  21233  coe1fzgsumdlem  22323  evl1gsumdlem  22376  pm2mp  22847  cayhamlem4  22910  iuninc  32581  cdlemk39s  40922  evl1gprodd  42099  minregex  43524
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