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Theorem csbing 3191
Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
Assertion
Ref Expression
csbing (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Proof of Theorem csbing
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2922 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥(𝐶𝐷))
2 csbeq1 2922 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
3 csbeq1 2922 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐷 = 𝐴 / 𝑥𝐷)
42, 3ineq12d 3186 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
51, 4eqeq12d 2097 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
6 vex 2615 . . 3 𝑦 ∈ V
7 nfcsb1v 2949 . . . 4 𝑥𝑦 / 𝑥𝐶
8 nfcsb1v 2949 . . . 4 𝑥𝑦 / 𝑥𝐷
97, 8nfin 3190 . . 3 𝑥(𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
10 csbeq1a 2927 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
11 csbeq1a 2927 . . . 4 (𝑥 = 𝑦𝐷 = 𝑦 / 𝑥𝐷)
1210, 11ineq12d 3186 . . 3 (𝑥 = 𝑦 → (𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷))
136, 9, 12csbief 2958 . 2 𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
145, 13vtoclg 2669 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  csb 2919  cin 2983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-in 2990
This theorem is referenced by:  csbresg  4672
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