Theorem difdif2 4271

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 4267	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
2		invdif 4254	. . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
3	2	eqcomi 2744	. . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶))
4	3	difeq2i 4098	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶)))
5		dfin2 4246	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶))
6	5	uneq2i 4140	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))
7	1, 4, 6	3eqtr4i 2768	1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶))

Syntax hints:   = wceq 1540  Vcvv 3459   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933

This theorem is referenced by:  psdmullem  22103  restmetu  24509  difelcarsg  34342  mblfinlem3  37683  mblfinlem4  37684