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Theorem csbing 3462
 Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
Assertion
Ref Expression
csbing (A B[A / x](CD) = ([A / x]C[A / x]D))

Proof of Theorem csbing
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3139 . . 3 (y = A[y / x](CD) = [A / x](CD))
2 csbeq1 3139 . . . 4 (y = A[y / x]C = [A / x]C)
3 csbeq1 3139 . . . 4 (y = A[y / x]D = [A / x]D)
42, 3ineq12d 3458 . . 3 (y = A → ([y / x]C[y / x]D) = ([A / x]C[A / x]D))
51, 4eqeq12d 2367 . 2 (y = A → ([y / x](CD) = ([y / x]C[y / x]D) ↔ [A / x](CD) = ([A / x]C[A / x]D)))
6 vex 2862 . . 3 y V
7 nfcsb1v 3168 . . . 4 x[y / x]C
8 nfcsb1v 3168 . . . 4 x[y / x]D
97, 8nfin 3230 . . 3 x([y / x]C[y / x]D)
10 csbeq1a 3144 . . . 4 (x = yC = [y / x]C)
11 csbeq1a 3144 . . . 4 (x = yD = [y / x]D)
1210, 11ineq12d 3458 . . 3 (x = y → (CD) = ([y / x]C[y / x]D))
136, 9, 12csbief 3177 . 2 [y / x](CD) = ([y / x]C[y / x]D)
145, 13vtoclg 2914 1 (A B[A / x](CD) = ([A / x]C[A / x]D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  [csb 3136   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by:  csbresg  4976
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