Theorem difindi 4215

Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

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

Proof of Theorem difindi

Step	Hyp	Ref	Expression
1		dfin3 4200	. . 3 ⊢ (𝐵 ∩ 𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
2	1	difeq2i 4054	. 2 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3		indi 4207	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4		dfin2 4194	. . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5		invdif 4202	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
6		invdif 4202	. . . 4 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
7	5, 6	uneq12i 4095	. . 3 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
8	3, 4, 7	3eqtr3i 2774	. 2 ⊢ (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))
9	2, 8	eqtri 2766	1 ⊢ (𝐴 ∖ (𝐵 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894

This theorem is referenced by:  difdif2  4220  indm  4222  fndifnfp  7048  dprddisj2  19642  fctop  22154  cctop  22156  mretopd  22243  restcld  22323  cfinfil  23044  csdfil  23045  indifundif  30873  difres  30939  unelcarsg  32279  clsk3nimkb  41650  ntrclskb  41679  ntrclsk3  41680  ntrclsk13  41681  salincl  43864  iscnrm3rlem1  46234