Theorem difdif2 4226

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

Assertion

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

Proof of Theorem difdif2

Step	Hyp	Ref	Expression
1		difindi 4221	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2		invdif 4208	. . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
3	2	eqcomi 2745	. . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶))
4	3	difeq2i 4060	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5		dfin2 4200	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶))
6	5	uneq2i 4100	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
7	1, 4, 6	3eqtr4i 2774	1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3437   ∖ cdif 3889   ∪ cun 3890   ∩ cin 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899

This theorem is referenced by:  restmetu  23775  difelcarsg  32326  mblfinlem3  35864  mblfinlem4  35865