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Theorem csbaovg 46589
Description: Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
csbaovg (𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )

Proof of Theorem csbaovg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3897 . . 3 (𝑦 = 𝐴𝑦 / 𝑥 ((𝐵𝐹𝐶)) = 𝐴 / 𝑥 ((𝐵𝐹𝐶)) )
2 csbeq1 3897 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3897 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3897 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4aoveq123d 46587 . . 3 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
61, 5eqeq12d 2744 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) ↔ 𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) ))
7 vex 3477 . . 3 𝑦 ∈ V
8 nfcsb1v 3919 . . . 4 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3919 . . . 4 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3919 . . . 4 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfaov 46588 . . 3 𝑥 ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
12 csbeq1a 3908 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3908 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3908 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14aoveq123d 46587 . . 3 (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) )
167, 11, 15csbief 3929 . 2 𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
176, 16vtoclg 3542 1 (𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  csb 3894   ((caov 46527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-aiota 46494  df-dfat 46528  df-afv 46529  df-aov 46530
This theorem is referenced by: (None)
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