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Theorem csbres 5854
Description: Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbres 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Proof of Theorem csbres
StepHypRef Expression
1 df-res 5563 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
21csbeq2i 3819 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
3 csbxp 5647 . . . . . 6 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
4 csbconstg 3830 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥V = V)
54xpeq2d 5581 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
63, 5eqtrid 2789 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
7 0xp 5646 . . . . . . 7 (∅ × V) = ∅
87a1i 11 . . . . . 6 𝐴 ∈ V → (∅ × V) = ∅)
9 csbprc 4321 . . . . . . 7 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
109xpeq1d 5580 . . . . . 6 𝐴 ∈ V → (𝐴 / 𝑥𝐶 × V) = (∅ × V))
11 csbprc 4321 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = ∅)
128, 10, 113eqtr4rd 2788 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
136, 12pm2.61i 185 . . . 4 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)
1413ineq2i 4124 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
15 csbin 4354 . . 3 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
16 df-res 5563 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
1714, 15, 163eqtr4i 2775 . 2 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
182, 17eqtri 2765 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wcel 2110  Vcvv 3408  csb 3811  cin 3865  c0 4237   × cxp 5549  cres 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5116  df-xp 5557  df-res 5563
This theorem is referenced by:  csbwrecsg  35235  csbima12gALTVD  42190
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