Theorem undir

Description: Distributive law for union over intersection. Theorem 29 of Suppes p. 27. (Contributed by NM, 30-Sep-2002)

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

Proof

Step	Hyp	Ref	Expression
1		undi	⊢ ( 𝐶 ∪ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐶 ∪ 𝐴 ) ∩ ( 𝐶 ∪ 𝐵 ) )
2		uncom	⊢ ( ( 𝐴 ∩ 𝐵 ) ∪ 𝐶 ) = ( 𝐶 ∪ ( 𝐴 ∩ 𝐵 ) )
3		uncom	⊢ ( 𝐴 ∪ 𝐶 ) = ( 𝐶 ∪ 𝐴 )
4		uncom	⊢ ( 𝐵 ∪ 𝐶 ) = ( 𝐶 ∪ 𝐵 )
5	3 4	ineq12i	⊢ ( ( 𝐴 ∪ 𝐶 ) ∩ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐶 ∪ 𝐴 ) ∩ ( 𝐶 ∪ 𝐵 ) )
6	1 2 5	3eqtr4i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∪ 𝐶 ) = ( ( 𝐴 ∪ 𝐶 ) ∩ ( 𝐵 ∪ 𝐶 ) )