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Theorem csbaovg 47192
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 3902 . . 3 (𝑦 = 𝐴𝑦 / 𝑥 ((𝐵𝐹𝐶)) = 𝐴 / 𝑥 ((𝐵𝐹𝐶)) )
2 csbeq1 3902 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3902 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3902 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4aoveq123d 47190 . . 3 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
61, 5eqeq12d 2753 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) ↔ 𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) ))
7 vex 3484 . . 3 𝑦 ∈ V
8 nfcsb1v 3923 . . . 4 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3923 . . . 4 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3923 . . . 4 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfaov 47191 . . 3 𝑥 ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
12 csbeq1a 3913 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3913 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3913 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14aoveq123d 47190 . . 3 (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) )
167, 11, 15csbief 3933 . 2 𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
176, 16vtoclg 3554 1 (𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  csb 3899   ((caov 47130
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-sep 5296  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-ne 2941  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-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132  df-aov 47133
This theorem is referenced by: (None)
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