Theorem difdif2 4285

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 4280	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2		invdif 4267	. . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
3	2	eqcomi 2741	. . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶))
4	3	difeq2i 4118	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5		dfin2 4259	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶))
6	5	uneq2i 4159	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
7	1, 4, 6	3eqtr4i 2770	1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1541  Vcvv 3474   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954

This theorem is referenced by:  restmetu  24070  difelcarsg  33297  mblfinlem3  36515  mblfinlem4  36516