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Theorem csbfv2g 6696
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 6695 . 2 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
2 csbconstg 3874 . . 3 (𝐴𝐶𝐴 / 𝑥𝐹 = 𝐹)
32fveq1d 6654 . 2 (𝐴𝐶 → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (𝐹𝐴 / 𝑥𝐵))
41, 3syl5eq 2869 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  csb 3855  cfv 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186  ax-pow 5243
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-dm 5542  df-iota 6293  df-fv 6342
This theorem is referenced by:  csbfv  6697  ixpsnval  8451  swrdspsleq  14018  sumeq2ii  15041  fsumabs  15147  prodeq2ii  15258  fprodabs  15319  ixpsnbasval  19973  coe1fzgsumdlem  20928  evl1gsumdlem  20978  pm2mp  21428  cayhamlem4  21491  iuninc  30319  cdlemk39s  38193
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