Theorem indir 1678

Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27.

Assertion

Ref	Expression
indir	|- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 1676	. 2 |- (C i^i (A u. B)) = ((C i^i A) u. (C i^i B))
2		incom 1636	. 2 |- ((A u. B) i^i C) = (C i^i (A u. B))
3		incom 1636	. . 3 |- (A i^i C) = (C i^i A)
4		incom 1636	. . 3 |- (B i^i C) = (C i^i B)
5	3, 4	uneq12i 1609	. 2 |- ((A i^i C) u. (B i^i C)) = ((C i^i A) u. (C i^i B))
6	1, 2, 5	3eqtr4 1126	1 |- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))

Syntax hints:   = wceq 1091   u. cun 1485   i^i cin 1486

This theorem is referenced by:  difundir 1682  undisj1 1739  resundir 2583

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491

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