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Theorem csbresgOLD 39555
Description: Distribute proper substitution through the restriction of a class. csbresgOLD 39555 is derived from the virtual deduction proof csbresgVD 39630. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5554 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 4153 . . 3 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
2 csbxp 5357 . . . . 5 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
3 csbconstg 3687 . . . . . 6 (𝐴𝑉𝐴 / 𝑥V = V)
43xpeq2d 5296 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
52, 4syl5eq 2806 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V))
65ineq2d 3957 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
71, 6syl5eq 2806 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
8 df-res 5278 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
98csbeq2i 4136 . 2 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))
10 df-res 5278 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
117, 9, 103eqtr4g 2819 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  csb 3674  cin 3714   × cxp 5264  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-in 3722  df-nul 4059  df-opab 4865  df-xp 5272  df-res 5278
This theorem is referenced by:  csbima12gALTVD  39632
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