Theorem undir 2257

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

Assertion

Ref	Expression
undir	⊢ ((A ∩ B) ∪ C) = ((A ∪ C) ∩ (B ∪ C))

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 2255	. 2 ⊢ (C ∪ (A ∩ B)) = ((C ∪ A) ∩ (C ∪ B))
2		uncom 2179	. 2 ⊢ ((A ∩ B) ∪ C) = (C ∪ (A ∩ B))
3		uncom 2179	. . 3 ⊢ (A ∪ C) = (C ∪ A)
4		uncom 2179	. . 3 ⊢ (B ∪ C) = (C ∪ B)
5	3, 4	ineq12i 2218	. 2 ⊢ ((A ∪ C) ∩ (B ∪ C)) = ((C ∪ A) ∩ (C ∪ B))
6	1, 2, 5	3eqtr4 1508	1 ⊢ ((A ∩ B) ∪ C) = ((A ∪ C) ∩ (B ∪ C))

Syntax hints:   = wceq 958   ∪ cun 2048   ∩ cin 2049

This theorem is referenced by:  undif1 2344

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054

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