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Theorem csbun 4437
Description: Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by NM, 13-Sep-2018.)
Assertion
Ref Expression
csbun 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Proof of Theorem csbun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3895 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵𝐶))
2 csbeq1 3895 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3895 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
42, 3uneq12d 4163 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
51, 4eqeq12d 2748 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
6 vex 3478 . . . 4 𝑦 ∈ V
7 nfcsb1v 3917 . . . . 5 𝑥𝑦 / 𝑥𝐵
8 nfcsb1v 3917 . . . . 5 𝑥𝑦 / 𝑥𝐶
97, 8nfun 4164 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
10 csbeq1a 3906 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3906 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1210, 11uneq12d 4163 . . . 4 (𝑥 = 𝑦 → (𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶))
136, 9, 12csbief 3927 . . 3 𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
145, 13vtoclg 3556 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
15 un0 4389 . . . 4 (∅ ∪ ∅) = ∅
1615a1i 11 . . 3 𝐴 ∈ V → (∅ ∪ ∅) = ∅)
17 csbprc 4405 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
18 csbprc 4405 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1917, 18uneq12d 4163 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅ ∪ ∅))
20 csbprc 4405 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = ∅)
2116, 19, 203eqtr4rd 2783 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2214, 21pm2.61i 182 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  Vcvv 3474  csb 3892  cun 3945  c0 4321
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-nul 4322
This theorem is referenced by:  csbprg  4712
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