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Theorem csbing 3288
 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 3010 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥(𝐶𝐷))
2 csbeq1 3010 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
3 csbeq1 3010 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐷 = 𝐴 / 𝑥𝐷)
42, 3ineq12d 3283 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
51, 4eqeq12d 2155 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
6 vex 2692 . . 3 𝑦 ∈ V
7 nfcsb1v 3040 . . . 4 𝑥𝑦 / 𝑥𝐶
8 nfcsb1v 3040 . . . 4 𝑥𝑦 / 𝑥𝐷
97, 8nfin 3287 . . 3 𝑥(𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
10 csbeq1a 3016 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
11 csbeq1a 3016 . . . 4 (𝑥 = 𝑦𝐷 = 𝑦 / 𝑥𝐷)
1210, 11ineq12d 3283 . . 3 (𝑥 = 𝑦 → (𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷))
136, 9, 12csbief 3049 . 2 𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
145, 13vtoclg 2749 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ⦋csb 3007   ∩ cin 3075 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-in 3082 This theorem is referenced by:  csbresg  4830
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