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Theorem csbaovg 40551
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 3522 . . 3 (𝑦 = 𝐴𝑦 / 𝑥 ((𝐵𝐹𝐶)) = 𝐴 / 𝑥 ((𝐵𝐹𝐶)) )
2 csbeq1 3522 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3522 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3522 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4aoveq123d 40549 . . 3 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
61, 5eqeq12d 2641 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) ↔ 𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) ))
7 vex 3194 . . 3 𝑦 ∈ V
8 nfcsb1v 3535 . . . 4 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3535 . . . 4 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3535 . . . 4 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfaov 40550 . . 3 𝑥 ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
12 csbeq1a 3528 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3528 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3528 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14aoveq123d 40549 . . 3 (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) )
167, 11, 15csbief 3544 . 2 𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
176, 16vtoclg 3257 1 (𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  csb 3519   ((caov 40486
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5813  df-fun 5852  df-fv 5858  df-dfat 40487  df-afv 40488  df-aov 40489
This theorem is referenced by: (None)
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