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Theorem csbdif 33482
Description: Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)
Assertion
Ref Expression
csbdif 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Proof of Theorem csbdif
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3677 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵𝐶))
2 csbeq1 3677 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3677 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
42, 3difeq12d 3872 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
51, 4eqeq12d 2775 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
6 vex 3343 . . . 4 𝑦 ∈ V
7 nfcsb1v 3690 . . . . 5 𝑥𝑦 / 𝑥𝐵
8 nfcsb1v 3690 . . . . 5 𝑥𝑦 / 𝑥𝐶
97, 8nfdif 3874 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
10 csbeq1a 3683 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3683 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1210, 11difeq12d 3872 . . . 4 (𝑥 = 𝑦 → (𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶))
136, 9, 12csbief 3699 . . 3 𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
145, 13vtoclg 3406 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
15 dif0 4093 . . . 4 (∅ ∖ ∅) = ∅
1615a1i 11 . . 3 𝐴 ∈ V → (∅ ∖ ∅) = ∅)
17 csbprc 4123 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
18 csbprc 4123 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1917, 18difeq12d 3872 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅ ∖ ∅))
20 csbprc 4123 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = ∅)
2116, 19, 203eqtr4rd 2805 . 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 1632  wcel 2139  Vcvv 3340  csb 3674  cdif 3712  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059
This theorem is referenced by: (None)
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