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Theorem csbing 3138
Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
Assertion
Ref Expression
csbing (A BA / x(𝐶𝐷) = (A / x𝐶A / x𝐷))

Proof of Theorem csbing
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2849 . . 3 (y = Ay / x(𝐶𝐷) = A / x(𝐶𝐷))
2 csbeq1 2849 . . . 4 (y = Ay / x𝐶 = A / x𝐶)
3 csbeq1 2849 . . . 4 (y = Ay / x𝐷 = A / x𝐷)
42, 3ineq12d 3133 . . 3 (y = A → (y / x𝐶y / x𝐷) = (A / x𝐶A / x𝐷))
51, 4eqeq12d 2051 . 2 (y = A → (y / x(𝐶𝐷) = (y / x𝐶y / x𝐷) ↔ A / x(𝐶𝐷) = (A / x𝐶A / x𝐷)))
6 vex 2554 . . 3 y V
7 nfcsb1v 2876 . . . 4 xy / x𝐶
8 nfcsb1v 2876 . . . 4 xy / x𝐷
97, 8nfin 3137 . . 3 x(y / x𝐶y / x𝐷)
10 csbeq1a 2854 . . . 4 (x = y𝐶 = y / x𝐶)
11 csbeq1a 2854 . . . 4 (x = y𝐷 = y / x𝐷)
1210, 11ineq12d 3133 . . 3 (x = y → (𝐶𝐷) = (y / x𝐶y / x𝐷))
136, 9, 12csbief 2885 . 2 y / x(𝐶𝐷) = (y / x𝐶y / x𝐷)
145, 13vtoclg 2607 1 (A BA / x(𝐶𝐷) = (A / x𝐶A / x𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  csb 2846  cin 2910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-in 2918
This theorem is referenced by:  csbresg  4558
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