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Theorem csbaovg 43723
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 3834 . . 3 (𝑦 = 𝐴𝑦 / 𝑥 ((𝐵𝐹𝐶)) = 𝐴 / 𝑥 ((𝐵𝐹𝐶)) )
2 csbeq1 3834 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3834 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3834 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4aoveq123d 43721 . . 3 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
61, 5eqeq12d 2817 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) ↔ 𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) ))
7 vex 3447 . . 3 𝑦 ∈ V
8 nfcsb1v 3855 . . . 4 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3855 . . . 4 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3855 . . . 4 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfaov 43722 . . 3 𝑥 ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
12 csbeq1a 3845 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3845 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3845 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14aoveq123d 43721 . . 3 (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) )
167, 11, 15csbief 3865 . 2 𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
176, 16vtoclg 3518 1 (𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  csb 3831   ((caov 43661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-aiota 43629  df-dfat 43662  df-afv 43663  df-aov 43664
This theorem is referenced by: (None)
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