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Theorem csbresgOLD 39370
Description: Distribute proper substitution through the restriction of a class. csbresgOLD 39370 is derived from the virtual deduction proof csbresgVD 39445. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5431 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbresgOLD (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem csbresgOLD
StepHypRef Expression
1 csbin 4043 . . 3 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
2 csbxp 5234 . . . . 5 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
3 csbconstg 3579 . . . . . 6 (𝐴𝑉𝐴 / 𝑥V = V)
43xpeq2d 5173 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
52, 4syl5eq 2697 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
65ineq2d 3847 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
71, 6syl5eq 2697 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
8 df-res 5155 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
98csbeq2i 4026 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
10 df-res 5155 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
117, 9, 103eqtr4g 2710 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  csb 3566  cin 3606   × cxp 5141  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-nul 3949  df-opab 4746  df-xp 5149  df-res 5155
This theorem is referenced by:  csbima12gALTVD  39447
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