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Theorem csbingOLD 38537
 Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 3982 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbingOLD (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Proof of Theorem csbingOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3517 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥(𝐶𝐷))
2 csbeq1 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
3 csbeq1 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐷 = 𝐴 / 𝑥𝐷)
42, 3ineq12d 3793 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
51, 4eqeq12d 2636 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
6 vex 3189 . . 3 𝑦 ∈ V
7 nfcsb1v 3530 . . . 4 𝑥𝑦 / 𝑥𝐶
8 nfcsb1v 3530 . . . 4 𝑥𝑦 / 𝑥𝐷
97, 8nfin 3798 . . 3 𝑥(𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
10 csbeq1a 3523 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
11 csbeq1a 3523 . . . 4 (𝑥 = 𝑦𝐷 = 𝑦 / 𝑥𝐷)
1210, 11ineq12d 3793 . . 3 (𝑥 = 𝑦 → (𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷))
136, 9, 12csbief 3539 . 2 𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
145, 13vtoclg 3252 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ⦋csb 3514   ∩ cin 3554 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 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  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 3418  df-csb 3515  df-in 3562 This theorem is referenced by:  csbresgOLD  38538  onfrALTlem5VD  38604  onfrALTlem4VD  38605  csbresgVD  38614
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