Theorem difdif2 4281

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 4276	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2		invdif 4263	. . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
3	2	eqcomi 2735	. . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶))
4	3	difeq2i 4114	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5		dfin2 4255	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶))
6	5	uneq2i 4155	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
7	1, 4, 6	3eqtr4i 2764	1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1533  Vcvv 3468   ∖ cdif 3940   ∪ cun 3941   ∩ cin 3942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950

This theorem is referenced by:  restmetu  24430  difelcarsg  33839  mblfinlem3  37038  mblfinlem4  37039