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Theorem csbres 5361
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 5088 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
21csbeq2i 3967 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
3 csbxp 5163 . . . . . 6 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
4 csbconstg 3528 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥V = V)
54xpeq2d 5101 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
63, 5syl5eq 2667 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
7 0xp 5162 . . . . . . 7 (∅ × V) = ∅
87a1i 11 . . . . . 6 𝐴 ∈ V → (∅ × V) = ∅)
9 csbprc 3954 . . . . . . 7 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
109xpeq1d 5100 . . . . . 6 𝐴 ∈ V → (𝐴 / 𝑥𝐶 × V) = (∅ × V))
11 csbprc 3954 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = ∅)
128, 10, 113eqtr4rd 2666 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
136, 12pm2.61i 176 . . . 4 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)
1413ineq2i 3791 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
15 csbin 3984 . . 3 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
16 df-res 5088 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
1714, 15, 163eqtr4i 2653 . 2 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
182, 17eqtri 2643 1 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  csb 3515  cin 3555  c0 3893   × cxp 5074  cres 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-opab 4676  df-xp 5082  df-res 5088
This theorem is referenced by:  csbwrecsg  32826
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