Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbaovg Structured version   Visualization version   GIF version

Theorem csbaovg 39909
 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 3502 . . 3 (𝑦 = 𝐴𝑦 / 𝑥 ((𝐵𝐹𝐶)) = 𝐴 / 𝑥 ((𝐵𝐹𝐶)) )
2 csbeq1 3502 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3502 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3502 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4aoveq123d 39907 . . 3 (𝑦 = 𝐴 → ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
61, 5eqeq12d 2625 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) ↔ 𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) ))
7 vex 3176 . . 3 𝑦 ∈ V
8 nfcsb1v 3515 . . . 4 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3515 . . . 4 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3515 . . . 4 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfaov 39908 . . 3 𝑥 ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
12 csbeq1a 3508 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3508 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3508 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14aoveq123d 39907 . . 3 (𝑥 = 𝑦 → ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)) )
167, 11, 15csbief 3524 . 2 𝑦 / 𝑥 ((𝐵𝐹𝐶)) = ((𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
176, 16vtoclg 3239 1 (𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⦋csb 3499   ((caov 39844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator