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Theorem csbafv12g 40518
Description: Move class substitution in and out of a function value, analogous to csbfv12 6188, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6640. (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 3517 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹'''𝐵) = 𝐴 / 𝑥(𝐹'''𝐵))
2 csbeq1 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afveq12d 40514 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2636 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵)))
6 vex 3189 . . 3 𝑦 ∈ V
7 nfcsb1v 3530 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3530 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv 40517 . . 3 𝑥(𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
10 csbeq1a 3523 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3523 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afveq12d 40514 . . 3 (𝑥 = 𝑦 → (𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3539 . 2 𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
145, 13vtoclg 3252 1 (𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  csb 3514  '''cafv 40495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-dfat 40497  df-afv 40498
This theorem is referenced by: (None)
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