Theorem difdif2

Description: Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017)

	Ref	Expression
Assertion	difdif2	⊢ ( 𝐴 ∖ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		difindi	⊢ ( 𝐴 ∖ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∖ ( V ∖ 𝐶 ) ) )
2		invdif	⊢ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) = ( 𝐵 ∖ 𝐶 )
3	2	eqcomi	⊢ ( 𝐵 ∖ 𝐶 ) = ( 𝐵 ∩ ( V ∖ 𝐶 ) )
4	3	difeq2i	⊢ ( 𝐴 ∖ ( 𝐵 ∖ 𝐶 ) ) = ( 𝐴 ∖ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) )
5		dfin2	⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐴 ∖ ( V ∖ 𝐶 ) )
6	5	uneq2i	⊢ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐶 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∖ ( V ∖ 𝐶 ) ) )
7	1 4 6	3eqtr4i	⊢ ( 𝐴 ∖ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐶 ) )