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Theorem csbfv 6956
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 6955 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴 / 𝑥𝑥))
2 csbvarg 4434 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
32fveq2d 6910 . . 3 (𝐴 ∈ V → (𝐹𝐴 / 𝑥𝑥) = (𝐹𝐴))
41, 3eqtrd 2777 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
5 csbprc 4409 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = ∅)
6 fvprc 6898 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
75, 6eqtr4d 2780 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴))
84, 7pm2.61i 182 1 𝐴 / 𝑥(𝐹𝑥) = (𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  c0 4333  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569
This theorem is referenced by:  mptcoe1fsupp  22217  mptcoe1matfsupp  22808  mp2pm2mplem4  22815  chfacfscmulfsupp  22865  chfacfpmmulfsupp  22869  cpmidpmatlem3  22878  cayhamlem4  22894  cayleyhamilton1  22898  logbmpt  26831  nbgrcl  29352  nbgrnvtx0  29356  iuninc  32573  disjxpin  32601  finixpnum  37612  cdlemkid3N  40935  cdlemkid4  40936  cdlemk39s  40941  mccllem  45612  clnbgrcl  47808  clnbgrnvtx0  47814
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