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Theorem csbin 4395
Description: Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
csbin 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Proof of Theorem csbin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3890 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵𝐶))
2 csbeq1 3890 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3890 . . . . 5 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
42, 3ineq12d 4194 . . . 4 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
51, 4eqeq12d 2842 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
6 vex 3503 . . . 4 𝑦 ∈ V
7 nfcsb1v 3911 . . . . 5 𝑥𝑦 / 𝑥𝐵
8 nfcsb1v 3911 . . . . 5 𝑥𝑦 / 𝑥𝐶
97, 8nfin 4197 . . . 4 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
10 csbeq1a 3901 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3901 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1210, 11ineq12d 4194 . . . 4 (𝑥 = 𝑦 → (𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶))
136, 9, 12csbief 3921 . . 3 𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
145, 13vtoclg 3573 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
15 csbprc 4362 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = ∅)
16 csbprc 4362 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
17 csbprc 4362 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1816, 17ineq12d 4194 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (∅ ∩ ∅))
19 in0 4349 . . . 4 (∅ ∩ ∅) = ∅
2018, 19syl6req 2878 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2115, 20eqtrd 2861 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2214, 21pm2.61i 183 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3500  csb 3887  cin 3939  c0 4295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-in 3947  df-nul 4296
This theorem is referenced by:  csbres  5855  disjxpin  30272  csbpredg  34495  onfrALTlem5  40760  onfrALTlem4  40761  onfrALTlem5VD  41103  onfrALTlem4VD  41104  csbresgVD  41113  disjinfi  41338
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