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Theorem csbing 3357
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 3075 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥(𝐶𝐷))
2 csbeq1 3075 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
3 csbeq1 3075 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐷 = 𝐴 / 𝑥𝐷)
42, 3ineq12d 3352 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
51, 4eqeq12d 2204 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
6 vex 2755 . . 3 𝑦 ∈ V
7 nfcsb1v 3105 . . . 4 𝑥𝑦 / 𝑥𝐶
8 nfcsb1v 3105 . . . 4 𝑥𝑦 / 𝑥𝐷
97, 8nfin 3356 . . 3 𝑥(𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
10 csbeq1a 3081 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
11 csbeq1a 3081 . . . 4 (𝑥 = 𝑦𝐷 = 𝑦 / 𝑥𝐷)
1210, 11ineq12d 3352 . . 3 (𝑥 = 𝑦 → (𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷))
136, 9, 12csbief 3116 . 2 𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
145, 13vtoclg 2812 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  csb 3072  cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-in 3150
This theorem is referenced by:  csbresg  4928
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