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Theorem csbres 5999
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 5696 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
21csbeq2i 3906 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
3 csbxp 5784 . . . . . 6 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
4 csbconstg 3917 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥V = V)
54xpeq2d 5714 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
63, 5eqtrid 2788 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
7 0xp 5783 . . . . . . 7 (∅ × V) = ∅
87a1i 11 . . . . . 6 𝐴 ∈ V → (∅ × V) = ∅)
9 csbprc 4408 . . . . . . 7 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
109xpeq1d 5713 . . . . . 6 𝐴 ∈ V → (𝐴 / 𝑥𝐶 × V) = (∅ × V))
11 csbprc 4408 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = ∅)
128, 10, 113eqtr4rd 2787 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
136, 12pm2.61i 182 . . . 4 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)
1413ineq2i 4216 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
15 csbin 4441 . . 3 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
16 df-res 5696 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
1714, 15, 163eqtr4i 2774 . 2 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
182, 17eqtri 2764 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2107  Vcvv 3479  csb 3898  cin 3949  c0 4332   × cxp 5682  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690  df-res 5696
This theorem is referenced by:  csbfrecsg  8310  csbima12gALTVD  44922
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