Theorem indifdir

Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011) (Revised by BTernaryTau, 14-Aug-2024)

	Ref	Expression
Assertion	indifdir	⊢ ( ( 𝐴 ∖ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∖ ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		indifdi	⊢ ( 𝐶 ∩ ( 𝐴 ∖ 𝐵 ) ) = ( ( 𝐶 ∩ 𝐴 ) ∖ ( 𝐶 ∩ 𝐵 ) )
2		incom	⊢ ( ( 𝐴 ∖ 𝐵 ) ∩ 𝐶 ) = ( 𝐶 ∩ ( 𝐴 ∖ 𝐵 ) )
3		incom	⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐶 ∩ 𝐴 )
4		incom	⊢ ( 𝐵 ∩ 𝐶 ) = ( 𝐶 ∩ 𝐵 )
5	3 4	difeq12i	⊢ ( ( 𝐴 ∩ 𝐶 ) ∖ ( 𝐵 ∩ 𝐶 ) ) = ( ( 𝐶 ∩ 𝐴 ) ∖ ( 𝐶 ∩ 𝐵 ) )
6	1 2 5	3eqtr4i	⊢ ( ( 𝐴 ∖ 𝐵 ) ∩ 𝐶 ) = ( ( 𝐴 ∩ 𝐶 ) ∖ ( 𝐵 ∩ 𝐶 ) )