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Theorem csbafv12g 39866
Description: Move class substitution in and out of a function value, analogous to csbfv12 6141, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6585. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
csbafv12g (𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Proof of Theorem csbafv12g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3502 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹'''𝐵) = 𝐴 / 𝑥(𝐹'''𝐵))
2 csbeq1 3502 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3502 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afveq12d 39862 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2625 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵)))
6 vex 3176 . . 3 𝑦 ∈ V
7 nfcsb1v 3515 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3515 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv 39865 . . 3 𝑥(𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
10 csbeq1a 3508 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3508 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afveq12d 39862 . . 3 (𝑥 = 𝑦 → (𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3524 . 2 𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
145, 13vtoclg 3239 1 (𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  csb 3499  '''cafv 39843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-dfat 39845  df-afv 39846
This theorem is referenced by: (None)
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