Theorem undir 2254

Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27.

Assertion

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

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 2252	. 2 |- (C u. (A i^i B)) = ((C u. A) i^i (C u. B))
2		uncom 2176	. 2 |- ((A i^i B) u. C) = (C u. (A i^i B))
3		uncom 2176	. . 3 |- (A u. C) = (C u. A)
4		uncom 2176	. . 3 |- (B u. C) = (C u. B)
5	3, 4	ineq12i 2215	. 2 |- ((A u. C) i^i (B u. C)) = ((C u. A) i^i (C u. B))
6	1, 2, 5	3eqtr4 1505	1 |- ((A i^i B) u. C) = ((A u. C) i^i (B u. C))

Syntax hints:   = wceq 956   u. cun 2045   i^i cin 2046

This theorem is referenced by:  undif1 2340

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051

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