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Theorem csbafv12g 41719
Description: Move class substitution in and out of a function value, analogous to csbfv12 6388, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6846. (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 3673 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹'''𝐵) = 𝐴 / 𝑥(𝐹'''𝐵))
2 csbeq1 3673 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3673 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afveq12d 41715 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2771 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵)))
6 vex 3339 . . 3 𝑦 ∈ V
7 nfcsb1v 3686 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3686 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv 41718 . . 3 𝑥(𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
10 csbeq1a 3679 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3679 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afveq12d 41715 . . 3 (𝑥 = 𝑦 → (𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3695 . 2 𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
145, 13vtoclg 3402 1 (𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1628  wcel 2135  csb 3670  '''cafv 41696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-res 5274  df-iota 6008  df-fun 6047  df-fv 6053  df-dfat 41698  df-afv 41699
This theorem is referenced by: (None)
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