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Theorem csbun 3981
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 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵𝐶))
2 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3517 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
42, 3uneq12d 3746 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
51, 4eqeq12d 2636 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
6 vex 3189 . . . 4 𝑦 ∈ V
7 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐵
8 nfcsb1v 3530 . . . . 5 𝑥𝑦 / 𝑥𝐶
97, 8nfun 3747 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
10 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3523 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1210, 11uneq12d 3746 . . . 4 (𝑥 = 𝑦 → (𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶))
136, 9, 12csbief 3539 . . 3 𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
145, 13vtoclg 3252 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
15 un0 3939 . . . 4 (∅ ∪ ∅) = ∅
1615a1i 11 . . 3 𝐴 ∈ V → (∅ ∪ ∅) = ∅)
17 csbprc 3952 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
18 csbprc 3952 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1917, 18uneq12d 3746 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅ ∪ ∅))
20 csbprc 3952 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = ∅)
2116, 19, 203eqtr4rd 2666 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2214, 21pm2.61i 176 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  csb 3514  cun 3553  c0 3891
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-nul 3892
This theorem is referenced by:  csbprg  4215
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