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Theorem csbfv 6395
Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem csbfv
StepHypRef Expression
1 csbfv2g 6394 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴 / 𝑥𝑥))
2 csbvarg 4146 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
32fveq2d 6357 . . 3 (𝐴 ∈ V → (𝐹𝐴 / 𝑥𝑥) = (𝐹𝐴))
41, 3eqtrd 2794 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
5 csbprc 4123 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = ∅)
6 fvprc 6347 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
75, 6eqtr4d 2797 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
84, 7pm2.61i 176 1 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  csb 3674  c0 4058  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057
This theorem is referenced by:  mptcoe1fsupp  19807  mptcoe1matfsupp  20829  mp2pm2mplem4  20836  chfacfscmulfsupp  20886  chfacfpmmulfsupp  20890  cpmidpmatlem3  20899  cayhamlem4  20915  cayleyhamilton1  20919  logbmpt  24746  nbgrcl  26447  nbgrclOLD  26448  nbgrnvtx0  26452  iuninc  29707  disjxpin  29729  cnfinltrel  33570  finixpnum  33725  cdlemkid3N  36741  cdlemkid4  36742  cdlemk39s  36747  mccllem  40350
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