MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbun Structured version   Visualization version   GIF version

Theorem csbun 4399
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 3859 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵𝐶))
2 csbeq1 3859 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3859 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
42, 3uneq12d 4125 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
51, 4eqeq12d 2749 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
6 vex 3448 . . . 4 𝑦 ∈ V
7 nfcsb1v 3881 . . . . 5 𝑥𝑦 / 𝑥𝐵
8 nfcsb1v 3881 . . . . 5 𝑥𝑦 / 𝑥𝐶
97, 8nfun 4126 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
10 csbeq1a 3870 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3870 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1210, 11uneq12d 4125 . . . 4 (𝑥 = 𝑦 → (𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶))
136, 9, 12csbief 3891 . . 3 𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
145, 13vtoclg 3524 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
15 un0 4351 . . . 4 (∅ ∪ ∅) = ∅
1615a1i 11 . . 3 𝐴 ∈ V → (∅ ∪ ∅) = ∅)
17 csbprc 4367 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
18 csbprc 4367 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1917, 18uneq12d 4125 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅ ∪ ∅))
20 csbprc 4367 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = ∅)
2116, 19, 203eqtr4rd 2784 . 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 1542  wcel 2107  Vcvv 3444  csb 3856  cun 3909  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-nul 4284
This theorem is referenced by:  csbprg  4671
  Copyright terms: Public domain W3C validator