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Theorem csbres 5985
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 5689 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
21csbeq2i 3902 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
3 csbxp 5776 . . . . . 6 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
4 csbconstg 3913 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥V = V)
54xpeq2d 5707 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
63, 5eqtrid 2782 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
7 0xp 5775 . . . . . . 7 (∅ × V) = ∅
87a1i 11 . . . . . 6 𝐴 ∈ V → (∅ × V) = ∅)
9 csbprc 4407 . . . . . . 7 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
109xpeq1d 5706 . . . . . 6 𝐴 ∈ V → (𝐴 / 𝑥𝐶 × V) = (∅ × V))
11 csbprc 4407 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = ∅)
128, 10, 113eqtr4rd 2781 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
136, 12pm2.61i 182 . . . 4 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)
1413ineq2i 4210 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
15 csbin 4440 . . 3 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
16 df-res 5689 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
1714, 15, 163eqtr4i 2768 . 2 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
182, 17eqtri 2758 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2104  Vcvv 3472  csb 3894  cin 3948  c0 4323   × cxp 5675  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-res 5689
This theorem is referenced by:  csbfrecsg  8273  csbima12gALTVD  43962
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